# A review of Alfred North Whitehead’s “Introduction to Mathematics”

Thomas Hales
University of Pittsburgh

##  1 An Introduction to Mathematics

In 1911, Alfred North Whitehead published a short book _Introduction to
Mathematics_ (IM) intended for students wanting an explanation of the
fundamental ideas of mathematics [14].

Whitehead’s IM has enduring value because it was written not long after he and
Bertrand Russell published their monumental three-volume work _Principia
Mathematica_ (PM) – a publication of immense historical significance for
mathematics [7]. IM sheds light on Whitehead’s view of mathematics at that
time. For Russell, the PM was the fulfillment of the logicist thesis: the
claim that all of mathematics is logic and nothing but logic. However, Russell
might have claimed more than PM could deliver, because the PM included the
axiom of choice and the axiom of infinity, which are not part of logic. The
primary components of their system were predicate logic, relations, and types.

Russell and Whitehead both turned away from mathematics after becoming
exhausted by the ten years they spent writing the PM. Russell wrote in his
autobiography, “My intellect never quite recovered from the strain. I have
been ever since definitely less capable of dealing with difficult abstractions
than I was before. This is part, though by no means the whole, of the reason
for the change in the nature of my work.” In 1910, Whitehead resigned his
Senior Lectureship at Trinity College, Cambridge, and moved to London, where
he was unemployed for a year [4]. His interests shifted from mathematics to
education, and IM was his first book after PM. IM is a thus a retrospective
and transitional work, written for the masses by a man recovering from a long
intellectual ordeal.

The book IM was published in the series “The Home University Library of Modern
Knowledge,” which published concise introductions on a broad range of topics
for a general audience. We might compare this series with similar book series
published today, such as the “Very Short Introduction,” “For Dummies,” and the
French “Que sais-je?” Bertrand Russell also published two books in the Home
University series, including his _The Problems of Philosophy_ [6]. (As an
aside, Russell’s _Problems_ was the topic of one of my college entrance essays
when I was a young man.)

Although a polymath, Whitehead was a mathematical physicist above all, and his
book IM contains a mixture of math and physics. The closest parallel to
Whitehead today might be Roger Penrose, whose books cover a broad range of
math, physics, and philosophy [5].

IM tells the famous story of Archimedes’ eureka moment of devising a test of
the purity of the gold in the king’s crown by submerging it in water to
measure its specific gravity. According to Whitehead, “This day, if we knew
which it was, ought to be celebrated as the birthday of mathematical physics.”
IM was written in the early days of quantum physics and relativity, but
Whitehead does not mention the new physics. Instead, IM recounts the
development of classical physics, including the fundamental forces of gravity,
electricity, and magnetism, and Newton’s laws of motion.

Vectors and linear algebra were not nearly so well known by math and
engineering students in 1910 as they are today. Whitehead’s IM helped to
popularize vectors as a tool of applied mathematics, especially in his
presentation of topics such as the parallelogram rule for the addition of
forces. Whitehead’s earlier treatise on universal algebra contains several
chapters on vectors, matrices, and forces. The treatise developed Cayley’s
theory of matrices and the properties of vectors as conceived by Grassmann and
Gibbs. According to Whitehead in IM, “The idea of the ‘vector’, that is, of a
directed magnitude, is the root-idea of physical science. … Thus, when in
analytical geometry the ideas of the ‘origin’, of ‘co-ordinates’, and of
‘vectors’ are introduced, we are studying the abstracts conceptions which
correspond to the fundamental facts of the physical world.” Vectors are used
to illustrate the main object of the book IM, which is to show students why
mathematics “is necessarily the foundation of exact thought as applied to
natural phenomena.”

In Whitehead’s technical writings, his meaning is sometimes obscure, because
of his insistence on generality. However, in his books written for a large
audience such as IM, there are many memorable sentences. One of the most
widely quoted sentences from IM foretold the rise of automation: “Civilization
advances by extending the number of important operations which we can perform
without thinking about them.” Another passage foreshadows the concepts of
entropy and compression in information theory: symbolism “should be concise,…
Now we cannot place symbols more concisely than by placing them in immediate
juxtaposition. In a good symbolism therefore, the juxtaposition of important
symbols should have an important meaning.” Whitehead saw the quest for the
permanent and the general as fundamental, “To see what is general in what is
particular and what is permanent in what is transitory is the aim of
scientific thought.… This possibility of disentangling the most complex
evanescent circumstances into various examples of permanent laws is the
controlling idea of modern thought.”

In IM, Whitehead tells the well-known story that Alexander the Great
impatiently asked for his geometry tutor to make the proofs shorter. His tutor
Menaechmus made the famous reply, “In the country there are private and even
royal roads, but in geometry there is only one road for all.” Whitehead
disagreed, “But if Menaechmus thought that his proofs could not be shorteneed,
he was grievously mistaken,… Nothing illustrates better the gain in power
which is obtained by the introduction of relevant ideas into a science than to
observe the progressive shortening of proofs which accompanies the growth of
richness in idea.… There are royal roads in science; but those who first tread
them are men of genius and not kings.”

##  2 Some and Any

Whitehead wrote in IM, “Mathematics as a science commenced when first someone,
probably a Greek, proved propositions about _any_ things or about _some_
things, without specification of definite particular things. When Whitehead
mentions the words _any_ and _some_ , he is referring to what is now called
predicate logic: the logic of the quantifiers _for all_ $(\forall)$ and _there
exists_ $(\exists)$. IM thus places proofs in predicate logic as the mythical
starting point of mathematics.

This statement in IM is particularly remarkable, because Whitehead himself was
slow to understand the significance of predicate logic. Russell learned of
symbolic notation for quantifiers from Peano and Frege, who had a major
influence on Russell starting in 1900, when Russell met Peano in Paris at the
International Congress of Mathematicians. It was Peano who introduced the
symbol $\exists$ for the existential quantifier, which was adopted in PM.
Russell and Whitehead used $(x)$ to mean “for all $x$,” and it was only later
that Hilbert introduced the symbol $(\forall)$.

Surprisingly, it appears that Whitehead failed to grasp the significance of
symbolic notation for quantifiers when he wrote _Treatise on Universal
Algebra_ (UA), which was published in 1898 [9]. UA developed algebra and
geometry from axioms with precise definitions, very much in the spirit of
Bourbaki a few decades later. The treatise is divided into books, and Book II
is _The Algebra of Symbolic Logic_. Within that book, Whitehead has an entire
chapter on _existential expressions_. From his citations in Book II, it is
known that Whitehead was well aware of the research on logic by Charles
Saunders Pierce and his students. Pierce had developed notation for
quantifiers by 1883 [2]. Whitehead also cited Schröder’s work, which
introduced Pierce’s ideas to Europe. However, it seems that Whitehead truly
failed to grasp the significance of authors he cited in UA until after
Russell’s encounter with Peano in 1900.

Whitehead endorsed the logicist thesis by writing, “A notable discovery has
been made by the joint and partially independent work of three men: Frege,
Peano and Bertrand Russell –a German, an Italian and an Englishman–… of the
generalized conception of the variable and of its essential presence in all
mathematical reasoning. This discovery empties mathematics of everything but
its logic” [13]. In this context, when Whitehead speaks of the variable, he
means the bound variable in logic. He explained explicitly in IM, “The idea of
the undetermined ‘variable’ as occurring in the use of ‘some’ or ‘any’ is the
really important one in mathematics.” Whitehead admitted in IM that a full
understanding of quantifiers had come only recently, “It was not till within
the last few years that it has been realized how fundamental _any_ and _some_
are to the very nature of mathematics.”

##  3 Relations and Axioms

As mentioned above, one of the main components of the logic of PM was a theory
of relations or predicates. In logic, a predicate is interpreted as a relation
of zero, one, or more variables. A function is a special kind of relation.
Over a period of many years, Whitehead developed various axiomatic systems of
geometry (and by extension, the natural world). Each axiomatic system can be
described as a collection of entities that are joined by relations that are
governed by axioms. If we try to summarize Whitehead’s life endeavor in a
single sentence, we might turn to a sentence in IM: “The events of this ever-
shifting world are but examples of a few general connexions or relations
called laws.”

Whitehead was part of the axiomatization movement in mathematics that became
widespread through Hilbert’s influence. In the ancient and medieval world, the
axiomatic reasoning of Euclid’s _Elements_ served as the paradigm. By the
nineteenth century, logic and mathematical rigor had advanced far beyond
Euclid’s ancient standards. Mathematicians such as Pasch and Peano started to
redo Euclid according to more rigorous standards. In 1899, Hilbert published
_Grundlagen der Geometrie (Foundations of Geometry)_ , which brought Euclid
into the twentieth century with a fresh set of definitions and axioms [3].

According to Eves, “By developing a postulate set for Euclidean geometry that
does not depart too greatly in spirit from Euclid’s own, and by employing a
minimum of symbolism, Hilbert succeeded in convincing mathematicians to a far
greater extent than had Pasch and Peano, of the purely hypothetico-deductive
nature of geometry. But the influence of Hilbert’s work went far beyond this,
for, backed by the author’s great mathematical authority, it firmly implanted
the postulational method, not only in the field of geometry, but also in
essentially every other branch of mathematics. The stimulus to the development
of the foundations of mathematics provided by Hilbert’s little book is
difficult to overestimate” [1].

Eventually, general foundations for mathematics were provided by the type
theory of PM (and also, independently by the Zermelo axioms of set theory).
These foundational systems have had an enormous influence on the growth of
mathematics. Before then, each branch of mathematics was separately
axiomatized. To axiomatize a domain of mathematics means to systematize it by
deciding which relations are to be considered as fundamental and by
enunciating a set of axioms from which the entire theory can be deduced.

During the decade that he and Russell devoted to the PM, as a participant in
the axiomatization movement, Whitehead wrote books on the axioms of geometry
and physics [12][10]. Whitehead’s aim in one publication during that period
was to generalize the axioms of static geometry to give a system of axioms
that describe the dynamical laws of physics [11]. Each system in that book
gives relations on a collection of undefined entities, subject to a list of
axioms.

Even his earlier treatise on universal algebra was a project in axiomatization
of abstract algebra. For example, the treatise contains the axioms for a
general nonassociative semiring. Theorems were derived from the definitions
and axioms in a style of abstract algebra that was brought into the mainstream
of mathematics decades later by B. L. van der Waerden in his book _Moderne
Algebra_ [8].

##  4 Whitehead’s Three Tests of Accuracy

How is an axiomatic mathematical theory to be judged? Whitehead suggested
three tests of accuracy: _“The tests of accuracy are logical coherence,
adequacy, and exemplification_ ”[15].111Whitehead proposed these tests to
evaluate metaphysical theories, but here the tests are applied to mathematical
theories. These words were written in 1926, several years before Gödel
published his incompletness theorems, but Whitehead’s three tests of accuracy
might be understood more fully in a post-Gödelian context. (Recall that
Gödel’s first version of the incompleteness theorem was written about the
mathematical system developed by Russell and Whitehead.) The first test of
accuracy is logical coherence, which might be taken to mean syntactic
consistency. According to Whitehead, only this first test is required to
establish validity of reasoning [9, p. vi]. Adequacy is a pragmatic test. In a
mathematical system, adequacy might mean that it contains sufficient
arithmetic to carry out the constructions of Gödel’s incompleteness theorem.
In a general metaphysical system, adequacy might mean adequacy for the purpose
of sustaining an advanced scientific civilization. Exemplification means that
the theory admits a model (in the sense of model theory). The first and third
tests, logical coherence and exemplification, are related by Henkin’s theorem
(1949): if a theory is syntactically consistent, then it has a model.

As suggested by this short review, Whitehead developed themes in IM that have
enduring value.

## References

  * Eves [1963] Howard Eves.  _A Survey of Geometry_ , volume I.  Allyn and Bacon, 1963.  quoted in https://en.wikipedia.org/wiki/Foundations_of_geometry.
  * Ewald [2019] William Ewald.  The Emergence of First-Order Logic.  In Edward N. Zalta, editor, _The Stanford Encyclopedia of Philosophy_. Metaphysics Research Lab, Stanford University, Spring 2019 edition, 2019.
  * Hilbert [1899] David Hilbert.  _Grundlagen der Geometrie_.  Springer-Verlag, 1899.
  * Lowe [1985, 1990] Victor Lowe.  _Alfred North Whitehead, The Man and His Work_ , volume I,II.  Johns Hopkins, 1985, 1990.
  * Penrose [2004] Roger Penrose.  _The Road to Reality_.  Jonathan Cape, 2004.
  * Russell [1912] Bertrand Russell.  _The Problems of Philosophy_.  Williams and Norgate, 1912.
  * Russell and Whitehead [1910, 1912, 1913] Bertrand Russell and Alfred N Whitehead.  _Principia Mathematica_.  Cambridge University Press, 1910, 1912, 1913.
  * van der Waerden [1930] B. L. van der Waerden.  _Moderne Algebra_.  Springer-Verlag, 1930.
  * Whitehead [1898] Alfred N Whitehead.  _A Treatise on Universal Algebra_.  Cambridge University Press, 1898.
  * Whitehead [1906a] Alfred N Whitehead.  _The Axioms of Projective Geometry_.  Cambridge Tracts in Mathematics and Mathematical Physics, 1906a.
  * Whitehead [1906b] Alfred N Whitehead.  On mathematical concepts of the material world.  _Philosophical transactions of the Royal Society of London, A_ , 205:465–525, 1906b.
  * Whitehead [1907] Alfred N Whitehead.  _The Axioms of Descriptive Geometry_.  Cambridge Tracts in Mathematics and Mathematical Physics, 1907.
  * Whitehead [1910] Alfred N Whitehead.  The philosophy of mathematics.  _Science progress in the twentieth century_ , 5:234–239, 1910.
  * Whitehead [1911] Alfred N Whitehead.  _An Introduction to Mathematics_.  Williams and Norgate, 1911.
  * Whitehead [1926] Alfred N Whitehead.  _Religion in the Making_.  Macmillan, 1926.


# Dataset for anomalies detection in 3D printing

Joanna Sendorek, Tomasz Szydlo, Mateusz Windak, Robert Brzoza-Woch  AGH
University of Science and Technology,
Department of Computer Science, Krakow, Poland
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Nowadays, Internet of Things plays a significant role in many domains.
Especially, Industry 4.0 is making a great usage of concepts like smart
sensors and big data analysis. IoT devices are commonly used to monitor
industry machines and detect anomalies in their work. In this paper we present
and describe a set of data streams coming from working 3D printer. Among
others, it contains accelerometer data of printer head, intrusion power and
temperatures of the printer elements. In order to gain data we lead to several
printing malfunctions applied to the 3D model. Resulting dataset can therefore
be used for anomalies detection research.

##  I Introduction

This paper presents the data that we have gathered from the 3D printer during
the printing process. Among all, data samples include temperature of working
elements of the printer, intrusion force and the acceleration of printing
head. The data has been gathered using two types of sources - custom-made
measurement devices and the internal software of the printer.

In order to enable for the dataset to serve as an example of anomalies
detection for intelligent Industry 4.0 systems, we provoked several types of
failures during printing process. All of the files are placed in the
repository111https://github.com/joanna-/3D-Printing-Data and can be used under
the Creative Commons Attribution 4.0 International license.

The rest of the paper is organised as follows. Section II presents related
work. In section III we describe the characteristics of printer machine used
for gathering data samples. In IV we characterize each type of data source
while printing failures that we created are presented in V. Section VI contain
specification of data organisation while section VII contains exemplary data
analysis. The last section sums up the paper.

##  II Related Work

As an interest in 3D printing increases in various applications, anomaly
detection systems are gaining in importance. We can basically distinguish two
types of systems used to monitor the work of 3D printers and received
printouts. These systems are based on image analysis (e.g. [1, 2]) and based
on the analysis of data from inertial sensors (e.g. [3, 4]).

Proper preparation of the 3D printer and retrofitting it with sensors requires
some time and equipment expenditure. In the article we present a set of test
data that were obtained using devices built using the FogDevices platform. The
presented set of test data can be used to develop new algorithms for detecting
anomalies in the work of 3D printers and, what is important, to compare them.

##  III 3D Printer characteristics

The 3D printer utilized for collecting its operation data was Monkeyfab Spire
manufactured by Monkeyfab222http://www.monkeyfab.com \- its basic properties
are listed in Table I. It is a _delta printer_ in which the printing head is
mounted on magnetic ball joints. The Monkeyfab Spire uses the _RepRapFirmware_
and is controlled over the network via _Duet Web Control_
333https://duet3d.dozuki.com/Wiki/Duet_Web_Control_Manual (access for
20.11.2019) interface.

TABLE I: Basic parameters of the utilized 3D printer
according to manufacturer’s specifications. Maximum printed object dimensions |  |  $150\text{\,}\mathrm{mm}$ diameter
---
$165\text{\,}\mathrm{mm}$ height
Default nozzle diameter | $0.4\text{\,}\mathrm{mm}$
Minimum layer height | $0.05\text{\,}\mathrm{mm}$
Filament diameter | $1.75\text{\,}\mathrm{mm}$
Maximum hotend temperature | $262\text{\,}\mathrm{\SIUnitSymbolCelsius}$
Maximum platform temperature | $120\text{\,}\mathrm{\SIUnitSymbolCelsius}$

##  IV Data sources characteristics

The sensor data comes from two sources - (i) internal electronics that control
the operation of the printer and from (ii) additionally mounted sensors. They
are described in more details in the next subsections.

###  IV-A Duet Web Control

Duet Web Control Interface is the user interface (UI), accessible via web
browser, that allows to monitor and change printer state. Among others, it
includes such features as: emergency stop, monitoring temperatures of printer
parts, changing filament and selecting 3D models to print. The aforementioned
informations are also exposed via API in json format which can be accessed at
[printer server address]/rr_status?type=X, where X is the category of data
format. In the created datasets, we have used two categories - 1 and 3
alternatively. The core data provided by the API is the same for these
categories, but they differ in some extra information. For example,
aforementioned third version includes currently printed layer of the 3D model.

###  IV-B Data acquisition hardware

The printer has been equipped with additional custom sensors developed as part
of the FogDevices444http://fogdevices.agh.edu.pl research project. Data from
them was collected using a device assembled using modular hardware components.

Figure 1: The utilized data acquisition system.

The printer has been equipped with two inertial measurement unit (IMU) sensors
LSM9DS1 that can measure acceleration, angular rate and magnetic field in 3
axis but only linear acceleration was used in this case. First of the sensors,
called _accel0_ is attached to the printing platform and _accel1_ is on the
print head. Both of the sensors use the I2C digital interface and are
connected to the FogDevices hardware platform.

The method of measuring the filament feeding force is based on indirect
measurement of the force acting on the Bowden tube during printer operation.
This was possible due to the fact that the extruder is located on the body of
the printer, not at the print head. Therefore, a force sensor _SG_ based on a
strain gauge was developed. Its operation is based on Wheatstone bridge and it
produces small voltage output. The voltage is amplified in FogDevices sensor
interface module with INA128 instrumentation amplifiers and then measured
using an analog-to-digital converter (ADC) with 12-bit resolution.

Block diagram of the hardware is presented in figure 1. The FogDevices
hardware platform has been utilized to collect data from three sensors: _SG_ ,
_accel0_ , and _accel1_.

Data collected by the device was being sent through the MQTT protocol over the
Ethernet interface. The data were then saved by a data logger running on a PC
computer. The acquisition system collects and processes 200 samples per
second.

Additional sensors and devices are provided by the FogDevices platform. The
video showing printing process is available
online555https://youtu.be/SFBInVsVDgk.

##  V Type of prints

We have used two variants of the same five towers print in order to collect
data. In the variant (a), presented in figures 2(a) and 2(c), towers have
printed base that is integral part of the print and in variant (b) presented
in figures 2(b) and 2(d), towers do not have a base - they are placed only on
the raft.

(a) towers with the base - schema

(b) towers without base - schema

(c) towers with the base - photo

(d) towers without base - photo

Figure 2: Printing schemas

For both variants, we have collected data from the undisturbed, properly made
print. Apart from that, we provoked six printing anomalies presented in figure
3:

  1. 1.

variant (a):

     * •

3(b) \- printer ran out of plastic before the print was finished;

     * •

3(c) \- part of the print unstuck from the printing base, but the rest of
print remained undisturbed;

     * •

3(d) \- speed of the retraction has been set too low (to 0.5);

     * •

3(e) \- during the printing, the Bowden tube fell out from its place;

     * •

3(f) \- during the printing, the arm of printer head has been detached from
magnets holding it in the place;

  2. 2.

variant (b):

     * •

3(a) \- during the printing, part of the print has been removed.

(a) removal of the part of the print

(b) plastic finish

(c) print unsticking

(d) 0.5 retraction

(e) Bowden tube fallout

(f) arm failure

Figure 3: Various malfunctions of the print

##  VI Data organization

All of the data is uploaded to the repository on
GitHub666https://github.com/joanna-/3D-Printing-Data. Structure of the
directories is as follows:

.1 3D-Printing-Data. .2 four_towers. .3 arm_failure. .3 bowden. .3 plastic. .3
proper. .3 retraction. .3 unstick. .3 towers_v01.stl. .2 four_towers_no_base.
.3 removing_part. .3 proper. .3 towers_only_v01.stl.

Each subdirectory contains zipped directory of the two files: csv data and
json data. Apart from that, four_towers and four_towers_no_base contain ṡtl
files with the printing schemas.

###  VI-A Custom measurements

Custom measurements are stored in files with csv format without header line
and with a standard comma separator. Each file contains the following columns:

  * •

0 \- data id assigned as the relative time set up on the device;

  * •

1-3 \- 3-dimensional data from the printing base (X, Y, Z axis respectively);

  * •

4-6 \- 3-dimensional data from the head accelerometer (X, Y, Z axis
respectively);

  * •

7 \- tension measurements;

  * •

8 \- time stamps (in milliseconds).

Listing 1: Csv data reading

⬇

import pandas as pd

df = pd.read_csv(’accel.txt’, names=[’data_id’, ’accel0X’, ’accel0Y’,
’accel0Z’, ’accel1X’, ’accel1Y’,’accel1Z’, ’tension’, ’timestamp’])

df[’time’] = pd.to_datetime(df[’timestamp’], unit=’ms’)

df[’tension’] = 0.650 * (df[’tension’] - 2166)

In Listing 1, we present suggested way of reading the data using pandas
library. The last line is needed as the effect of measurement device
calibration - applying given formula scales the filament force data to start
from 0.

###  VI-B Measurements from Duet Web Control

Data that is provided by the web interface is in json file, in which each line
contain different data sample object. The object includes many different key-
values pairs and we will not discuss all of them in this paper as it is
described in the device documentation. Below, we present several ones, that
were useful during our analysis:

  * •

coords.xyz \- array of length three with current (x, y, z) position of
printing head;

  * •

status \- can be one of the following values:

    1. 1.

I \- for idle state;

    2. 2.

P \- for printing phase;

    3. 3.

T \- for temporary when printer is getting ready to printing phase;

  * •

temps.bed.current \- current temperature of the bed;

  * •

temps.extra \- among all contain MCU temperature;

  * •

currentLayer* \- current printed layer - available only in some datasets.

In Listing 2, we present suggested way of reading the json data using pandas
library. Normalization of json data results in the flat (not nested) table
structure for the obtained data.

Listing 2: Json data reading

⬇

import pandas as pd

from pandas.io.json import json_normalize

df = pd.read_json(’interface.json’, lines=True)

df = json_normalize(df.to_dict(’records’))

##  VII Exemplary Data analysis

failure type | symptoms | brief explanation
---|---|---
finish of plastic |  decrease of intrusion power | there is no more plastic to intrude
Bowden tube fallout |  | there is no friction with the print - plastic
---
doesn’t reach printed model
wrong retraction (0.5) | printing base jolting | too much plastic hooks on the next layers
unsticking of the model | printing head hooks on the rolled print
arm failure | printing head angle change | detachment of arm causes head to tilt
TABLE II: Symptoms characteristic of the printing failures.

Provoked failures cause different symptoms that can be detected with the data
analysis. Different failures may have similar symptoms depending on their type
and therefore inferring the initial cause can require more complex analysis.
In this section we present very basics of analysis and show three types of
symptoms related to five types of failures. The summary of failure-symptom
correlation is presented in the table II.

Figure 4 presents two different plots that show some of the aforementioned
symptoms. Figure 4a shows the situation where the filament feeding force
dropped rapidly at time 11:40. That symptom may suggest that the filament is
over or there was severe mechanical problem - in this case the Bowden tube
fallout. Figure 4b shows the tilt angle of the print head during printing.
Values different from 180 degrees are caused by the fact that the angle is
calculated on the basis of the accelerometer placed on the head, which is
affected by the force of gravity and acceleration resulting from the movement
of the head during printing. At 11:00 a significant change in the value of the
graph can be observed on the chart indicating mechanical damage to the
printer. In this case, the arm fixing the printing head in the delta system is
damaged.

The presented analysis is only an example of using data to analyze the work of
a 3D printer.

(a) Tension values for the print with Bowden tube fallout.

(b) Tilt angle values for the print with head arm detachment.

Figure 4: Plots presenting symptoms of printing anomalies

## Summary

The article presents the possibilities offered by the use of IoT devices in
industry 4.0. Retrofitting machines with additional sensors and devices
analyzing their work in real-time can provide valuable information about their
work.

IoT devices such as those offered by FogDevices Platform allow to simplify the
process of adding sensors and analyzing data on the edge, near the sensors
without sending them to the computational clouds.

The article presents data collected during the operation of the 3D printer,
including typical errors. The collected data can be used to develop advanced
algorithms for detection and prediction of failures.

## Data usage

The dataset is under Creative Commons Attribution 4.0 International license.
Please cite this paper if you use it.

## Acknowledgment

The research presented in this paper was supported by the National Centre for
Research and Development (NCBiR) under Grant No.
LIDER/15/0144/L-7/15/NCBR/2016.

## References

  * [1] G. Wu, Z. Shen, X. Shang, H. Wu, G. Xiong, and J. Yang, “3d printer optical detection system based on dlp projection technology,” in _2019 Chinese Automation Congress (CAC)_ , 2019, pp. 1040–1045.
  * [2] Baumann, Felix and Roller, Dieter, “Vision based error detection for 3d printing processes,” _MATEC Web of Conferences_ , vol. 59, p. 06003, 2016\. [Online]. Available: https://doi.org/10.1051/matecconf/20165906003
  * [3] C. Kim, D. Espalin, A. Cuaron, M. A. Perez, E. MacDonald, and R. B. Wicker, “A study to detect a material deposition status in fused deposition modeling technology,” in _2015 IEEE International Conference on Advanced Intelligent Mechatronics (AIM)_ , 2015, pp. 779–783.
  * [4] J. Windau and L. Itti, “Inertial machine monitoring system for automated failure detection,” in _2018 IEEE International Conference on Robotics and Automation (ICRA)_ , 2018, pp. 93–98.


# A solution of fermion sign problem for large fermion systems

Yunuo Xiong  College of Science, Zhejiang University of Technology, Hangzhou
31023, China  Hongwei Xiong<EMAIL_ADDRESS>College of Science, Zhejiang
University of Technology, Hangzhou 31023, China

###### Abstract

The exchange antisymmetry between identical fermions gives rise to infamous
fermion sign problem, e.g. both path integral Monte Carlo and path integral
molecular dynamics could only give accurate result for only a few
noninteracting fermions at low temperatures. By considering fictitious
particles with a real parameter interpolating continuously between bosons and
fermions, we use path integral molecular dynamics to propose a general
strategy to solve fermion sign problem in polynomial time. We verify that our
method can efficiently give accurate energy values for large fermion systems
through a series of numerical experiments.

The partition function is the starting point of quantum statistical physics.
Based on Feynman’s idea of path integral feynman ; kleinert ; Tuckerman , we
can transform the partition function of many-particle Fermi systems into
classical interacting ring polymers chandler ; Parrinello ; Haber ; Thomas
and use Monte Carlo Dornheim ; DornheimMod  or molecular dynamics (MD)
HirshbergFermi ; Xiong2  approaches to sample the probability distribution in
the partition function to obtain physical quantities such as the average
energy. However, due to the exchange antisymmetry of fermions, the
distribution function in the partition function is not positive definite,
rendering existing sampling methods ineffective. Despite the development of
various approaches to alleviate the fermion sign problem DornheimMod ;
HirshbergFermi , there are still insurmountable difficulties for large fermion
systems at low temperatures Alex ; ceperley ; troyer ; loh ; lyubartsev ; vozn
. For a long time it is widely believed that fermion sign problem is a NP-hard
problem troyer .

In this work (see Fig. 1 and Fig. 2 for a brief summary of our method and
idea), we notice that the partition function $Z(\xi)$ of a fictitious quantum
system is an analytical function of a real variable $\xi$ with $\xi=-1$ for
fermions, $\xi=0$ for distinguishable particles, and $\xi=1$ for bosons. By
using a recursion formula HirshbergFermi ; Hirshberg ; Deuterium ; Xiong ;
Xiong2  for identical particles, we can accurately and efficiently calculate
the average energy $E(\xi)$ for $\xi\geq 0$, without suffering from the
fermion sign problem. Because both $Z(\xi)$ and $E(\xi)$ are analytical
function of $\xi$, we find that the analytical continuation to fermions of
$\xi=-1$ can give us accurate energy for fermions with the computational
scaling of $O(N^{3})$, which provides excellent strategy to solve the fermion
sign problem.

We used numerical experiments to verify our idea with large fermion systems
greatly exceeding the computational power of traditional methods in path
integral formalism. For example, in our numerical experiments, we obtain the
predicted results accurately even for a system consisting of 20 fermions
without interactions. Traditional method cannot obtain the same result even
with millions times more computational resources. Needless to say, the
strategy for fermion sign problem we propose here will produce novel ways to
solve significant problems in many disciplines such as physics, chemistry and
astronomy.

Figure 1:  (a) Illustration of all the ring polymer configurations for 2
identical particles to calculate the partition function. $\xi$ denotes the
exchange effect between particles. (b) shows roughly the ratio of
computational steps between the traditional method and our method to consider
the fermion sign problem. The big difference is due to the computational
scaling of $O(e^{\beta N})$ in the traditional method and $O(PN^{3})$ in our
method. As shown in (c), in our method, we first calculate the energy
$E(\beta,\xi)$ for different positive $\xi$ without suffering from fermion
sign problem. Then, the fitting with parabolic function is used to get the
energy of 4 interacting fermions with $\beta=1$ and $\lambda=0.5$. The red
circle gives the energy of the traditional method Dornheim . (d) shows perfect
agreement between parabolic function and the traditional method in the whole
region $-1\leq\xi\leq 1$ for 4 interacting fermions. In our simulations, 12
beads and 3 independent trajectories each consisting of $5\times 10^{6}$ MD
steps are used to assure convergence.

Partition function of identical particles and fermion sign problem—For the
following Hamiltonian operator of $N$ identical particles

$\hat{H}=\frac{1}{2m}\sum_{l=1}^{N}\hat{\textbf{p}}_{l}^{2}+\hat{V}(\textbf{r}_{1},\cdots,\textbf{r}_{N}),$
(1)

the partition function $Z(\beta,\xi)=Tr(e^{-\beta\hat{H}})$ is given by

$Z(\beta,\xi)=\frac{1}{N!}\sum_{p\in S_{N}}\xi^{N_{p}}\int
d\textbf{r}_{1}d\textbf{r}_{2}\cdots
d\textbf{r}_{N}\left<p\\{\textbf{r}\\}|e^{-\beta\hat{H}}|\\{\textbf{r}\\}\right>.$
(2)

Here $\beta=1/k_{B}T$, with $k_{B}$ being the Boltzmann constant and $T$ being
the system temperature. $S_{N}$ represents the set of $N!$ permutation
operations denoted by $p$. The factor $\xi^{N_{p}}$ is due to the exchange
effect of identical particles, with $N_{p}$ a number defined to be the minimum
number of times for which pairs of indices must be interchanged to recover the
original order. $\xi=+1$ for boson partition function, while $\xi=-1$ for
fermion partition function. In addition, $\\{\textbf{r}\\}$ denotes
$\\{\textbf{r}_{1},\cdots,\textbf{r}_{N}\\}$. Using
$e^{-\beta\hat{H}}=e^{-\Delta\beta\hat{H}}\cdots e^{-\Delta\beta\hat{H}}$ with
$\Delta\beta=\beta/P$ and the technique of path integral, the partition
function $Z(\beta,\xi)$ can be mapped as a classical system of interacting
ring polymers chandler ; Parrinello ; Haber ; Thomas . See Fig. 1(a) as an
example for two identical particles.

In recent works by Hirshberg et al. Hirshberg ; HirshbergFermi , a recursion
formula is found to calculate the partition function for both bosons ($\xi=1$)
and fermions ($\xi=-1$). Based on the path integral ring polymer chandler ;
Parrinello ; Haber ; Thomas , the fermion partition function ($Z_{F}$) and
boson partition function ($Z_{B}$) are expressed as an analytical function of
the coordinates of interacting ring polymers

$Z_{F/B}\sim\int d\textbf{R}_{1}\cdots d\textbf{R}_{N}e^{-\beta
U_{F/B}^{(N)}}.$ (3)

Here $Z_{F/B}$ becomes a function of a set of ring polymer coordinates
$(\textbf{R}_{1},\cdots,\textbf{R}_{N})$, with
$\textbf{R}_{i}=(\textbf{r}_{i}^{1},\cdots,\textbf{r}_{i}^{P})$ corresponding
to $P$ ring polymer coordinates for the $i$th particle. The recursion formula
is found to calculate $U_{F}^{(N)}$ for fermions and $U_{B}^{(N)}$ for bosons.
For bosons, $U_{B}^{(N)}$ is always real and $e^{-\beta U_{B}^{(N)}}$ may be
sampled for large boson systems Hirshberg ; Deuterium ; Xiong ; Xiong2 .
However, because of the factor $(-1)^{N_{p}}$ for fermions, $U_{F}^{(N)}$ is a
complex function, which can not be sampled with available methods.

Based on the traditional method DornheimMod ; HirshbergFermi  to deal with the
fermion sign problem, the fermion partition function may be written as
HirshbergFermi

$Z_{F}\sim\int d\textbf{R}_{1}\cdots d\textbf{R}_{N}e^{-\beta
U_{B}^{(N)}}e^{\beta(U_{B}^{(N)}-U_{F}^{(N)})}.$ (4)

In this case, we may still use molecular dynamics to sample $e^{-\beta
U_{B}^{(N)}}$, while the energy is $E(\beta)=\left<\epsilon
s\right>_{B}/{\left<s\right>_{B}}$. Here
$s=e^{\beta(U_{B}^{(N)}-U_{F}^{(N)})}$ and $\epsilon$ is the estimator for
energy. The average $\left<\cdots\right>_{B}$ is about the samples based on
the partition function $Z_{B}$ for bosons.

However, because of the factor $s$, it is extremely hard to calculate the
energy, both in path integral Monte Carlo ceperley ; Alex ; Science ; CeperRMP
; boninsegni1 ; boninsegni2 ; Dornheim ; DornheimMod  and path integral
molecular dynamics HirshbergFermi ; Xiong2 . General consideration shows that
the simulation becomes exponentially hard to converge with increasing numbers
of fermions and decreasing temperatures ceperley . This is known as the
infamous fermion sign problem ceperley ; Alex ; loh ; troyer ; lyubartsev ;
vozn ; Wu ; Umrigar ; Li ; Wei , which is the biggest obstacle for almost all
first-principle calculations for large fermion systems. It was declared by
Troyer and Wiese troyer  that an algorithm that converges in polynomial
time—is “almost certainly unattainable”. In this work, however, we will show
that we have effective strategies to solve the fermion sign problem with
polynomial time for a wide range of large fermion systems. Fig. 1(b) shows
roughly the ratio of computational steps between the traditional method and
our method to consider the fermion sign problem.

Figure 2:  For the partition function $Z(\beta,\xi)$ with $\xi$ a real number
($\xi=-1$ for fermions, $\xi=0$ for distinguishable particles and $\xi=1$ for
bosons), $U_{\xi}^{(N)}$ in the whole region $\xi\geq 0$ is a real function,
which can be sampled accurately and efficiently. Hence, we can get many
reliable values for the energy $E(\beta,\xi)$ in this region, and analytically
continue to the energy of fermions ( illustrated by blue solid line with
arrow) if $E(\beta,\xi)$ has simple analytical property about $\xi$. In the
traditional method, however, it is equivalent that the distribution of the
partition function for a single point $\xi=1$ is used to try to extend to the
energy for fermions illustrated by red dashed line, which leads to an illusion
of NP-hard problem.

Partition function for a fictitious quantum system and analytical
continuation— Because the operator $e^{-\beta\hat{H}}$ is hermitian, it is
clear that $Z(\beta,\xi)$ is a real function of $\beta$ and $\xi$ even for
$\xi=-1$. For fermions, the transformation of $Z(\beta,\xi)$ as a multiple
integral expression by path integral ring polymer is an important merit in the
traditional method. However, after this transformation, for $\xi=-1$
(fermions), $U_{F}^{(N)}$ in the exponential function $e^{-\beta U_{F}^{(N)}}$
in the partition function becomes a complex function. This is a huge cost to
solve fermions by path integral ring polymer. However, we should never forget
that $Z(\beta,\xi)$ in essence is a problem of real number, and the emergence
of complex exponential function and the resulting fermion sign problem is due
to specific mathematical technique.

Fortunately, even for integral about complex exponential function, we still
have mathematical method to solve the problem. A typical example is the
calculation of $\int_{-\infty}^{\infty}e^{iax^{2}}dx$ for a real number $a$.
If we use the usual numerical method to calculate this integral, we may have
the illusion that this is a NP-hard problem. The method of analytical
continuation makes this integral easy. If we try to consider directly the
integral in $Z(\beta,\xi=-1)$, it would be a NP-hard problem too troyer .
However, this does not mean that the calculation of the energy of identical
fermions must be a NP-hard problem.

To calculate the energy for identical fermions, we turn to consider the
partition function $Z(\beta,\xi)$ given by Eq. (2). From the expression of
$Z(\beta,\xi)$, it is clear that it is mathematically legitimate that we
extend two discrete values $\xi=\\{-1,1\\}$ for $Z(\beta,\xi)$ to the general
case that $\xi$ becomes a continuous real variable (See Fig. 2). Generalizing
the technique of the recursion formula discovered by Hirshberg et al.
Hirshberg ; HirshbergFermi , from Eq. (2), we have

$Z(\beta,\xi)\sim\int d\textbf{R}_{1}\cdots d\textbf{R}_{N}e^{-\beta
U_{\xi}^{(N)}}.$ (5)

The recursion formula of $U_{\xi}^{(N)}$ is given in SM (Supplementary
Material). The energy of this fictitious quantum system for general real
number $\xi$ is $E(\beta,\xi)=-\partial\ln Z(\beta,\xi)/{\partial\beta}$. It
is obvious that both $Z(\beta,\xi)$ and $E(\beta,\xi)$ are analytical
functions of $\beta$ and $\xi$.

From the recursion formula, $U_{\xi}^{(N)}$ is a real function for $\xi\geq
0$, while a complex function for $\xi<0$ (See Fig. 2). Therefore, in the whole
real number region $\xi\geq 0$, the partition function for this fictitious
quantum system does not suffer from fermion sign problem. Because of the
analytical characteristic of $Z(\beta,\xi)$ and $E(\beta,\xi)$ about $\beta$
and $\xi$, it is expected that the analytical continuation will give us good
chance to deal with fermions with $\xi=-1$. We consider in this paper the
usual polynomial fitting so that
$E(\beta,\xi)=a_{0}+a_{1}\xi+a_{2}\xi^{2}+\cdots a_{n}\xi^{n}+\cdots$. The
accurate calculation of $E(\beta,\xi)$ for $\xi\geq 0$ means that we can
obtain accurate evaluation of the coefficients $a_{j}$, in particular for
analytical function $E(\beta,\xi)$ about $\xi$. In this case, it’s a matter of
course that we have good chance to get accurate result of $E(\beta,\xi=-1)$
for fermions.

Nevertheless, for such seemingly natural idea, we must carry out numerical
experiments to put it to the ultimate test. We have used many different
fermion systems to justify our discoveries. The evaluation of the precision of
$E(\beta,\xi=-1)$ can be shown through the degree of conformity of all the
data for $\xi\geq 0$ in the fitting.

Numerical experiments for noninteracting and interacting fermions— Now we turn
to consider numerical experiments to test our method. In our numerical
simulations (more details are given in SI), the path integral ring polymers
and molecular dynamics with separate Nosé-Hoover thermostat Nose1 ; Nose2 ;
Hoover ; Martyna ; Jang  are used to sample the partition function
$Z(\beta,\xi)$ for $\xi\geq 0$, which is further used by polynomial fitting to
calculate the energy for fermions. The number of beads $P$ used decreases as
temperature increases to ensure numerical stability and assure convergence. In
all of the following we checked convergence with respect to the number of
beads and MD steps performed. For details of how to assure the convergence,
one may refer to the supplementary material in Ref. Hirshberg

We consider fermions in a two-dimensional harmonic trap
$\frac{1}{2}m\omega^{2}(x^{2}+y^{2})$ with the units of
$\hbar=1,\omega=1,m=1$. The interparticle interactions are
$V_{int}=\sum_{l<j}^{N}\lambda/{|{\textbf{r}}_{l}-{\textbf{r}}_{j}|}$. For
$N=4,\beta=1$ and $\lambda=0.5$ (not difficult both in traditional method and
our method), the blue solid circle in Fig. 1(c) shows the numerical result of
the energy for $0\leq\xi\leq 1$. The solid line is the best fitting with the
parabolic function. The red solid circle is the result of path integral Monte
Carlo in Ref. Deuterium . In Fig. 1(d), we give the numerical results for the
whole region $-1\leq\xi\leq 1$ by using the traditional method to deal with
the fermion sign problem. We give the parabolic function in the whole region,
and perfect agreement is found. This verifies that $E(\beta,\xi)$ is a simple
analytical function about $\xi$ in the whole region connecting bosons and
fermions.

Figure 3:  Left figure for N=6: The blue circle and blue line give the results
for $\beta=1$ in our method. The red circle gives the result of traditional
method Dornheim , while the green circle gives the result of grand canonical
ensemble. Right figure for N=20: The blue circle and blue line give the
results for $\beta=1$ by polynomial fitting, while the red circle is the
result of grand canonical ensemble. The inset shows that the parabolic
function fails in the fitting. Hence, we should include the term $\xi^{4}$ for
polynomial fitting.

We consider further an example of noninteracting fermions with $N=6$ and
$\beta=1$, which is very difficult but feasible in traditional method. It is
worth pointing out that the noninteracting fermions at low temperatures have
the most severe fermion sign problem, which provide the best system to check
the method to solve the fermion sign problemDornheimMod ; HirshbergFermi . In
Fig. 3(a), we give the numerical results for these parameters. The red circle
shows the result of traditional method by path integral Monte Carlo Dornheim .

Figure 4:  Left figure for N=8: The blue circle and blue line give our
results, while the red circle is the result of traditional method Dornheim .
Right figure for N=20: The blue circle and blue line give our results for
$\beta=1$ and $\lambda=0.5$. The inset in (b) shows the linear fit, which has
obvious derivation.

In Fig. 3(b), we consider noninteracting fermions of $20$ particles in two-
dimensional harmonic trap at $\beta=1$, which is far beyond the capabilities
of traditional method of path integral ring polymer. It is also much beyond
the capabilities of the improved method to alleviate the fermion sign problem
DornheimMod ; HirshbergFermi  by introducing an extra fictitious interaction.
We show our numerical result for $0\leq\xi\leq 2$ and the fitting to get the
energy for fermions. The energy of fermions agrees perfectly with the result
of grand canonical ensemble. Generally speaking, the actual interacting
fermion systems have less severe fermion sign problem, compared with ideal
fermions. Hence, the test of the principle of ideal fermions gives us strong
support to simulate efficiently and accurately large interacting fermion
systems.

Although ideal fermions have already provided the best and harshest example to
test our method, as the final example, we consider interacting fermions in
two-dimensional harmonic trap. In Fig. 4(a), we show our result for $N=8$,
$\beta=1$ and $\lambda=0.5$, which agrees with the traditional method (red
circle) by Dornheim Dornheim , while our method is much more efficient and has
much smaller statistical fluctuations. In Fig. 4(b), we show the numerical
simulation of $N=20$, $\beta=1$ and $\lambda=0.5$. The successful application
of our method to large fermion systems is not so surprising, because the
partition function $Z$ and total energy $E$ are also regular functions of $N$.
This means that the success of our method for several examples of different
$N$ will ensure the validity for all $N$ in principle, with computational
scaling of $O(PN^{3})$.

It is not the purpose of the present work to prove or argue that our
polynomial-time algorithm can be used to study any large fermion system. The
present method relies on the assumption that $E(\beta,\xi)$ has simple
analytical property for $\xi\geq-1$. In SM, we provide other examples to
suggest strongly that the simple analytical property is a very common
situation. The simple analytical property of $E(\beta,\xi)$ has physical
reason that the exchange effect of fermions has equivalently repulsive
statistical interaction, while the exchange effect of bosons leads to
equivalently attractive statistical interaction. This suggests strongly that
$E(\beta,\xi)$ has monotonic or simple behavior, for the vast majority of
quantum systems we have interests. Of course, for some specific quantum
systems, we may find better function than polynomial function for data
fitting. It depends on the quantum system studied, and the comparison with
experiments may also improve the choice of the fitting function.

In all the results based on our method, the statistical error is negligible.
Hence, we do not give error bar for the energy in our method. This provides
strict criteria to test whether our method is applicable for specific and
eccentric quantum systems in future work. In practical application, one may
first check our method with ideal fermions to avoid potential problem, and
then use it to interacting fermions (see details in SM).

Conclusion—Compared with the traditional method, at first sight, our method
and result for solving the fermion sign problem is an unbelievable miracle.
However, the miracle is due to the fact that in our method, more informations
(the reliable importance sampling for many different values of $\xi\geq 0$ for
$Z(\beta,\xi)$) are given to the analytical property of $Z(\beta,\xi)$ to
extend the result to $\xi=-1$ (See Fig. 2). As a comparison, we see that in
the traditional method, we only consider the distribution of the partition
function $Z(\beta,\xi)$ for a single point $\xi=1$ and try to analytically
extend to the result to $\xi=-1$. This traditional method for extrapolation is
obviously hard and dangerous. Based on this comparison, the superiority of our
method is self-evident. It is expected that the present work will stimulate
new surges of scientific research in many fields such as physics, chemistry
and astronomy.

###### Acknowledgements.

This work is partly supported by the National Natural Science Foundation of
China under grant numbers 11175246, and 11334001. We acknowledge useful
suggestion to improve the paper by X. S. Chen, X. Z Wang and X. Q. Lin. We
acknowledge in particular the encouragement and important comment to improve
the paper by Frank Wilczek.

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Supplementary Material for ”A solution of fermion sign problem for large
fermion systems”

##  Appendix A Methods

(I) Recursion formula for $Z(\beta,\xi)$

We consider $N$ fictitious identical particles, so that the normalized
position eigenstates are

$\left|N_{\xi}\right>=\frac{1}{\sqrt{N!}}\sum_{p\in
S_{N}}\xi^{N_{p}}\left|p\\{\textbf{r}\\}\right>.$ (A.1)

$S_{N}$ represents the set of $N!$ permutation operations denoted by $p$. The
factor $\xi^{N_{p}}$ is due to the exchange effect of the fictitious
particles, with $N_{p}$ a number defined to be the minimum number of times for
which pairs of indices must be interchanged to recover the original order.

We have the unit operator

$\hat{I}_{\xi}=\frac{1}{N!}\int d\textbf{r}_{1}d\textbf{r}_{2}\cdots
d\textbf{r}_{N}\left|N_{\xi}\right>\left<N_{\xi}\right|.$ (A.1)

We also have another unit operator

$\hat{I}=\int d\textbf{r}_{1}d\textbf{r}_{2}\cdots
d\textbf{r}_{N}\left|\\{\textbf{r}\\}\right>\left<\\{\textbf{r}\\}\right|.$
(A.1)

By dividing $\beta$ into $P$ segments, with $\Delta\beta=\beta/P$, and using
the identity

$e^{-\beta\hat{H}}\hat{I}_{\xi}=e^{-\Delta\beta\hat{H}}\hat{I}e^{-\Delta\beta\hat{H}}\hat{I}\cdots\hat{I}e^{-\Delta\beta\hat{H}}\hat{I}_{\xi},$
(A.1)

we can obtain the discrete form for the partition function by inserting the
definition for the identity operators from the above equations. After that is
done, the partition function is a function of a set of ring polymer
coordinates $(\textbf{R}_{1},\cdots,\textbf{R}_{N})$, with
$\textbf{R}_{i}=(\textbf{r}_{i}^{1},\cdots,\textbf{r}_{i}^{P})$ corresponding
to $P$ ring polymer coordinates for the $i$th particle. Explicitly, after
considering all the permutation terms in Eq. (2) in the text, we have
$U_{\xi}^{(N)}$ in Eq. (5) in the text given by

$U_{\xi}^{(N)}=-\frac{1}{\beta}\ln
W_{\xi}^{(N)}+\frac{1}{P}\sum_{j=1}^{P}V\left(\textbf{r}_{1}^{j},\cdots,\textbf{r}_{N}^{j}\right),$
(A.1)

and $W_{\xi}^{(N)}$ is

$W_{\xi}^{(N)}=\frac{1}{N}\sum_{k=1}^{N}\xi^{k-1}e^{-\beta
E_{N}^{(k)}}W_{\xi}^{(N-k)}.$ (A.1)

In addition,

$E_{N}^{(k)}=\frac{1}{2}m\omega_{P}^{2}\sum_{l=N-k+1}^{N}\sum_{j=1}^{P}\left(\textbf{r}_{l}^{j+1}-\textbf{r}_{l}^{j}\right)^{2}.$
(A.1)

Here $\textbf{r}_{l}^{P+1}=\textbf{r}_{l+1}^{1}$, except for $l=N$ for which
$\textbf{r}_{N}^{P+1}=\textbf{r}_{N-k+1}^{1}$. In addition,
$\omega_{P}=\sqrt{P}/\beta\hbar$.

(II) Energy estimator and molecular dynamics

In order to perform molecular dynamics, we need the potential function and its
gradient. From the fact that $W_{\xi}^{(N)}=e^{-\beta V_{\xi}^{(N)}}$, we can
evaluate the potential $V_{\xi}^{(N)}$ recursively as

$V_{\xi}^{(\alpha)}=-\frac{1}{\beta}\log{\frac{1}{\alpha}\sum_{k=1}^{\alpha}\xi^{k-1}e^{-\beta(E_{\alpha}^{(k)}+V_{\xi}^{(\alpha-k)})}}.$
(A.1)

Taking gradient, we see that the forces are given by

$-\nabla_{\mathbf{r}_{l}^{j}}V_{\xi}^{(\alpha)}=-\frac{\sum_{k=1}^{\alpha}\xi^{k-1}[\nabla_{\mathbf{r}_{l}^{j}}E_{\alpha}^{(k)}+\nabla_{\mathbf{r}_{l}^{j}}V_{\xi}^{(\alpha-k)}]e^{-\beta(E_{\alpha}^{(k)}+V_{\xi}^{(\alpha-k)})}}{\sum_{k=1}^{\alpha}\xi^{k-1}e^{-\beta(E_{\alpha}^{(k)}+V_{\xi}^{(\alpha-k)})}}.$
(A.1)

From the relation between $E(\beta,\xi)$ and $Z(\beta,\xi)$, we have

$E(\beta,\xi)=\frac{PdN}{2\beta}+\frac{\left<U\right>}{P}+\left<V_{\xi}^{(N)}+\beta\frac{\partial
V_{\xi}^{(N)}}{\partial\beta}\right>,$ (A.1)

with $d$ the spatial dimensions. $V_{\xi}^{(N)}+\beta\frac{\partial
V_{\xi}^{(N)}}{\partial\beta}$ may be evaluated as

$V_{\xi}^{(N)}+\beta\frac{\partial
V_{\xi}^{(N)}}{\partial\beta}=\frac{\sum_{k=1}^{N}\xi^{k-1}[V_{\xi}^{(N-k)}+\beta\frac{\partial
V_{\xi}^{(N-k)}}{\partial\beta}-E_{N}^{(k)}]e^{-\beta(E_{N}^{(k)}+V_{\xi}^{(N-k)})}}{\sum_{k=1}^{N}\xi^{k-1}e^{-\beta(E_{N}^{(k)}+V_{\xi}^{(N-k)})}},$
(A.1)

with $V_{\xi}^{(0)}+\beta\frac{\partial V_{\xi}^{(0)}}{\partial\beta}=0$.

We should take care to ensure numerical stability when evaluating the above
formulas. For the potential, except for the case of $\xi=0$, a numerically
stable way to calculate potential is the following

$V_{\xi}^{(\alpha)}=\frac{1}{\beta}(\tilde{E}-\log{\sum_{k=1}^{\alpha}e^{(k-1)\log{\xi}-\beta(E_{\alpha}^{(k)}+V_{\xi}^{(\alpha-k)})+\tilde{E}}}+\log{\alpha}),$
(A.1)

where $\tilde{E}$ is chosen as

$\tilde{E}=-\max_{k}[(k-1)\log{\xi}-\beta(E_{\alpha}^{(k)}+V_{\xi}^{(\alpha-k)})].$
(A.1)

(III) Polynomial fitting and applicability of our approach

Because in our method, the energy for $\xi\geq 0$ can be calculated accurately
(it is routine to realize the precision far below $1\%$), the only potential
problem to get the accurate value for fermions is whether the fitting is
reasonable. The good fitting depends on the good analytical property of
$E(\beta,\xi)$ between $-1\leq\xi\leq\xi_{up}$. Here $\xi_{up}$ is a positive
value in our numerical calculations between $0\leq\xi\leq\xi_{up}$. In this
work, we use Mathematica for fitting, while the energy between
$0\leq\xi\leq\xi_{up}$ is calculated by path integral molecular dynamics with
C++ codes. If $E(\beta,\xi)$ has singular behavior or highly irregular
oscillating behavior about $\xi$, it is hard to assure the extrapolation for
fermions. However, even for these situations, some unique methods such as
Schlessinger point method may be helpful. Fortunately, the exchange effect of
fermions has equivalently repulsive statistical interaction, while the
exchange effect of bosons leads to equivalently attractive statistical
interaction. This suggests strongly that $E(\beta,\xi)$ has simple monotonic
behavior, for most quantum systems we have interests. Of course, for any
practical research, we should first check the numerical stability and accuracy
for $\xi\geq 0$ and judge the validity of the fitting also from physical
considerations. Before considering large fermion systems, one may first
calculate a few fermions for the region $-1\leq\xi\leq\xi_{up}$ to confirm the
simple and monotonic behavior of $E(\beta,\xi)$, like Fig. 1(d) in this paper.
After this confirmation, it would be safe to get the energy for large fermion
systems using the data fitting for $\xi\geq 0$.

##  Appendix B Analytical property of $E(\beta,\xi)$ for other examples

As discussed in the text, the applicability of our method relies on the good
analytical property of $E(\beta,\xi)$. In this supplementary material, we
calculate $N=4,\beta=1$ for different types of interparticle interaction by
traditional method to deal with the fermion sign problem, to show good
analytical property of $E(\beta,\xi)$ for the whole region of $-1\leq\xi\leq
1$. Of course, for any practical research, we suggest strongly that one
confirms first the analytical property of $E(\beta,\xi)$ for $\xi\geq-1$ for a
few fermions by path integral molecular dynamics and traditional method to
deal with the fermion sign problem before the calculations for large fermion
systems with our method. See the general strategy to numerically simulate
large fermion systems in Fig. S.1.

Figure S.1:  The general strategy to safely simulate large fermion systems
without suffering from fermion sign problem.

In all the following simulations, we use parabolic function for fitting, and
the good analytical property of $E(\beta,\xi)$ about $\xi$ is shown very
clearly.

(1) Fig. S.2: Coulomb repulsive interaction
$V_{int}=\sum_{l<j}^{N}\frac{\lambda}{{|{\textbf{r}}_{l}-{\textbf{r}}_{j}|}}$
with $\lambda=1$.

Figure S.2:  Coulomb repulsive interaction.

(2) Fig. S.3: Dipole interaction
$V_{int}=\sum_{l<j}^{N}\frac{\lambda}{{|{\textbf{r}}_{l}-{\textbf{r}}_{j}|^{3}}}$
with $\lambda=1$.

Figure S.3:  Dipole interaction.

(3) Fig. S.4: Gaussian interaction $V_{int}=\sum_{l<j}^{N}\frac{g}{\pi
s^{2}}e^{-\frac{|{\textbf{r}}_{l}-{\textbf{r}}_{j}|^{2}}{s^{2}}}$ with $g=3.0$
and $s=0.5$.

Figure S.4:  Gaussian interaction.


# Aharonov–Bohm scattering for relativistic particles in (3 + 1)-dimensional
noncommutative space with spin dependence

C. A. Stechhahn  Faculdades Integradas Campos Salles, Rua Nossa Senhora da
Lapa, 270/284 – Lapa – CEP 05072-000 – São Paulo, SP, Brazil

###### Abstract

We study the effects of noncommutativity of spacetime with mixed spatial and
spin degrees of freedom in a relativistic context. Using the Dirac equation in
$(3+1)$ dimensions and in a symmetric gauge, we calculate the invariant
amplitude for a small magnetic field flux. The parameter $\theta$ that
characterizes the noncommutativity here is not constant, and the model
preserves Lorentz symmetry. A comparison is made with scattering in the
context of canonical noncommutativity.

##  I Introduction

In recent years, several aspects of noncommutative quantum theories have been
investigated (see Refs. RevModPhys.73.977,  for a review). In these studies,
noncommutativity is considered to be a fundamental description of spacetime at
the Planck scale CommMathPhys.172.187  or as a particular limit of a quantum
theory of gravity.JHEP.09.032

Noncommutative coordinates were first proposed as long ago as 1947 in an
attempt to remove the ultraviolet divergences that plague perturbative quantum
field theories,PhysRev.71.38  although related ideas were discussed even
earlier than that. NuclPhys.B108.30  However, the idea was abandoned after the
great success of the renormalization program.

With the advent of string theory, new interest in noncommutative (NC) space
emerged. PhysRev.D84.045002 ; PhysRev.D86.105035 ; JPhys.A.46.245303 ;
ModPhysLett.A.29.1450048 ; EurPhysJ.C74.3101 ; PhysLett.A378.3509 ;
PhysLett.B743.184 ; PhysLett.B761.462 ; JHEP.01.049 ; PhysLett.B766.181 ;
PhysLett.A382.3317 ; EurPhysJ.C79.300 ; UkrJPhys.64.131  It has been shown
that the dynamics of open strings in the presence of an antisymmetric
$B$-field can be described in certain limits by a gauge theory on a
noncommutative spacetime, JHEP.09.032  namely, one deformed by the Moyal
product (whose definition will be given shortly).

One well-known physical example of noncommutativity of coordinates appears in
quantum mechanics. The Landau problem PhysLett.B718.1475  in which a charged
particle moves in a strong constant magnetic field that is perpendicular to
the plane of motion leads to a noncommutative space when it is projected to a
lower Landau level.

At distances of the order of the Planck length, owing to the generation of
intense gravitational fields, the measurement of coordinates loses all
meaning, as does the concept of “point.” This suggests the introduction of
position operators that do not commute:

$[\hat{x}_{\mu},\hat{x}_{\nu}]=i\theta_{\mu\nu},$ (1)

where $\theta_{\mu\nu}$ is a constant antisymmetric matrix of dimension
(length)2 and the hatted quantities are position operators in NC space. The
nonlocality inherent to Eq. (1) leads to the appearance of several significant
effects, such as ultraviolet/infrared (UV/IR) mixing. JHEP.03.035  In this
context, another interesting aspect of NC spacetime is the violation of
unitarity and causality when $\theta_{0i}\neq 0$. NuclPhys.B591.265 ;
PhysLett.B533.178 ; EurPhysJ.C25.469 ; JHEP10.2004.072 ; JHEP.06.044 ;
JHEP.05.040 ; PhysRev.D69.065008  However, in all these situations,
$\theta_{\mu\nu}$ is constant, i.e, we have canonical noncommutativity. It is
therefore appropriate to consider novel cases where, for example, the
noncommutativity is spin-dependent PhysLett.B680.384 . We should emphasize
here that the NC Aharonov–Bohm (AB) effect has been studied in both the
quantum mechanical PhysRev.D76.085008  and field theory PhysRev.D70.085005 ;
PhysRev.D71.107701  contexts. More general situations may also be considered
where $\theta_{\mu\nu}$ is position-dependent, i.e., a new operator. In this
sense, Eq. (1) may be taken as a first approximation to a more general setting
where the commutator of coordinates itself is not a constant operator.
AnnPhys.260.224 ; PhysLett.B.555.83 ; EurPhysJ.54.325 ; PhysRev.D79.125011  In
Refs. PhysLett.B718.1475,  and JPhys.A.MathTheor.43.285301, , the physical
meaning of spin noncommutativity is discussed in terms of a nonstandard
Heisenberg algebra.

In the present work, we shall consider a relativistic quantum mechanical model
where the spin dependence is incorporated into the Pauli–Lubanski vector. We
calculate the scattering amplitude for relativistic particles in a symmetric
gauge. The aim of this work is to study this novel kind of noncommutativity
and its physical implications in contrast to the canonical case.

##  II The Aharonov–Bohm effect: symmetric gauge and spin noncommutativity

We consider the AB effect, with an infinitely thin solenoid lying along the
$z$ axis in an NC space and with coordinates $\hat{x}^{\mu}$ satisfying the
algebra

$\left[\hat{x}^{\mu},\hat{x}^{\nu}\right]=-i\theta\epsilon^{\mu\nu\rho\sigma}S_{\rho\sigma}+i\frac{\theta^{2}}{2}\epsilon^{\mu\nu\rho\sigma}W_{\rho}p_{\sigma},$
(2)

where

$W^{\mu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}S_{\rho\sigma}=\frac{1}{2}\gamma^{5}\sigma^{\mu\nu}\partial_{\nu},$
(3)

is the Pauli–Lubanski pseudovector, $S_{\rho\sigma}=\sigma_{\rho\sigma}/2$ is
the spin operator, and
$\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$. Whenever necessary,
we assume the Pauli representation for the gamma matrices:

$\gamma^{0}=\begin{pmatrix}\mathbf{I}&0\\\
0&-\mathbf{I}\end{pmatrix},\qquad\gamma^{i}=\begin{pmatrix}0&\sigma^{i}\\\
-\sigma^{i}&0\end{pmatrix},$ (4)

where the spacetime metric is $\eta^{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$ and
$\sigma^{i}$ are the Pauli matrices. The boost and rotation generators are
AnIntroductiontoQuantumFieldTheory.41

$S^{0i}=\frac{i}{4}[\gamma^{0},\gamma^{i}],\qquad
S^{ij}=\frac{i}{4}[\gamma^{i},\gamma^{i}],$ (5)

so the antisymmetric $S^{ij}=\sigma^{ij}/2$ and
$\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$. The algebra in Eq. (2)
can be implemented through the following definitions:

$\displaystyle x^{\mu}$
$\displaystyle\rightarrow\hat{x}^{\mu}=x^{\mu}\mathbf{1}+\theta W^{\mu},$ (6)
$\displaystyle p^{\mu}$ $\displaystyle\rightarrow\hat{p}^{\mu}=p^{\mu},$

where $x^{\mu}$ and $p^{\mu}$ satisfy the usual Heisenberg algebra.

We write the NC Dirac equation in a form that ensures a Hermitian Lagrangian
density:PhysLett.B718.1475

$\left\\{-i\gamma^{\mu}\partial_{\mu}+M+\frac{e}{2}\left[A_{\mu}(x)\gamma^{\mu}\star+A_{\mu}(x)\star\gamma^{\mu}\right]\right\\}\psi(x)=0,$
(7)

where the “star operation” (Moyal product) represents the following action of
a Weyl-ordered operator $f(\hat{x}^{\mu})$ on a spinor $\psi(x)$:

$f(\hat{x}^{\mu})\star\psi=f(x^{\mu}\mathbf{1}+\theta\gamma^{5}\sigma^{\mu\nu}\partial_{\nu})\psi=f(x^{\mu})\exp\\!\left(\theta\overleftarrow{\partial}_{\mu}\gamma^{5}\sigma^{\mu\nu}\overrightarrow{\partial}_{{}_{\nu}}\right)\psi.$
(8)

We adopt the symmetric ordering shown in Eq. (7) because the operator
$A_{\mu}(\hat{x}^{\mu})$ has a matrix structure that does not commute with
$\gamma^{\mu}$. In the relativistic case, the NC Dirac equation describes a
spin-1/2 particle in an external electromagnetic field in the presence of spin
noncommutativity.

In ordinary relativistic quantum mechanics, the Dirac equation for a particle
of mass $M$ and charge $-e$ in an external magnetic field reads

$\left[\alpha_{i}\left(p_{i}-eA_{i}\right)+\beta
M\right]\psi(j,x)=i\partial_{t}\psi(j,x),$ (9)

where $\alpha_{i}=\gamma^{0}\gamma^{i}$ and $\beta=\gamma^{0}$. The magnetic
field is derived from a vector potential $(A^{0}=A^{3}=0)$ such that

$eA_{i}=-\frac{e\Phi}{2\pi}\frac{\epsilon_{ij}x_{j}}{\rho^{2}}=-\alpha\frac{\epsilon_{ij}x_{j}}{\rho^{2}},$
(10)

where $\Phi$ is the magnetic flux, $\epsilon_{ij}$ is the Levi-Civita symbol
(normalized as $\epsilon_{12}=1$), and $\rho=\sqrt{x_{1}^{2}+x_{2}^{2}}$ .
This is the gauge potential (symmetric gauge) in an ordinary commutative
space, and it corresponds to $\vec{B}=\Phi\delta(\rho)\hat{k}$.

Thus, after performing the star product in the NC Dirac equation in the AB
potential, up to first order in $\theta$, Eq. (7) reads

$\left[-i\gamma^{0}\gamma^{i}\partial_{i}+M\gamma^{0}+e\gamma^{0}\gamma^{i}A_{i}(x)+\frac{e\theta}{2}\left(\partial_{k}A_{i}\right)\gamma^{0}\left\\{\gamma^{i},\gamma^{5}\sigma^{kj}\right\\}\partial_{j}\right]\psi(x)=i\partial_{t}\psi(x).$
(11)

We are using natural units, such that $\hbar=c=1$, and $e<0$ for the electron
charge. A straightforward calculation in the symmetric gauge (10) leads to the
total Hamiltonian $H$ as the sum of

$\displaystyle H_{0}$
$\displaystyle=-i\gamma^{0}\gamma^{i}\partial_{i}+M\gamma^{0}-\frac{e\Phi}{2\pi\rho^{2}}\epsilon_{ij}x_{j}\gamma^{0}\gamma^{i}$
(12) and $\displaystyle H_{\mathrm{int}}$
$\displaystyle=\frac{e\theta}{2}[\left(\partial_{1}A_{i}\right)\gamma^{0}\left\\{\gamma^{i},\gamma^{5}\sigma^{12}\right\\}\partial_{2}+\left(\partial_{2}A_{i}\right)\gamma^{0}\left\\{\gamma^{i},\gamma^{5}\sigma^{21}\right\\}\partial_{1}]\psi=i\partial_{t}\psi.$
(13)

After all the $\gamma$ products have been carried out and the $x_{i}$ and
$\partial_{i}$ terms have been properly grouped, $H_{\mathrm{int}}$ reduces to

$\displaystyle H_{\mathrm{int}}={}$
$\displaystyle\alpha\theta\left\\{\frac{2x_{1}x_{2}}{\rho^{4}}\begin{pmatrix}-i\sigma^{2}&0\\\
0&-i\sigma^{2}\end{pmatrix}\partial_{2}+\left(\frac{1}{\rho^{2}}-\frac{2x_{1}^{2}}{\rho^{4}}\right)\begin{pmatrix}i\sigma^{1}&0\\\
0&i\sigma^{1}\end{pmatrix}\partial_{2}\right.$ $\displaystyle\hskip
16.0pt\left.+\left(\frac{1}{\rho^{2}}-\frac{2x_{2}^{2}}{\rho^{4}}\right)\begin{pmatrix}-i\sigma^{2}&0\\\
0&-i\sigma^{2}\end{pmatrix}\partial_{1}+\frac{2x_{1}x_{2}}{\rho^{4}}\begin{pmatrix}i\sigma^{1}&0\\\
0&i\sigma^{1}\end{pmatrix}\partial_{1}\right\\}$ (14)

(for details, see Appendices B and C). After some algebra, this interaction
Hamiltonian may be written as

$H_{\mathrm{int}}(\rho,\varphi)=-\frac{\alpha\theta}{\rho^{3}}\begin{pmatrix}0&e^{-i\varphi}z^{+}_{ij}&0&0\\\
e^{i\varphi}z^{-}_{ij}&0&0&0\\\ 0&0&0&e^{-i\varphi}z^{+}_{ij}\\\
0&0&e^{i\varphi}z^{-}_{ij}&0\end{pmatrix},$ (15)

where

$z^{\pm}_{ij}=\left(i\epsilon_{ij}\pm\delta_{ij}\right)x_{i}\partial_{j}.$
(16)

The Dirac spinor solution of the Dirac equation in an AB potential, Eq. (9),
may be written as PhysRev.D53.2178

$\psi(j,x)=\frac{1}{2\pi}\frac{1}{\sqrt{2E_{p}}}e^{-iE_{p}t+ip_{3}z}e^{im\varphi}e^{i\pi|\ell|/2}\begin{pmatrix}\chi\\\\[3.0pt]
\zeta\end{pmatrix},$ (17)

where $m$ is the integer part of the parameter $\alpha=e\Phi/2\pi$ in Eq.
(10): $\alpha=m+\delta$, with $0<\delta<1$. The two-component eigenvectors are
given by

$\displaystyle\chi$
$\displaystyle=\frac{1}{\sqrt{2s}}\begin{pmatrix}\sqrt{E_{p}+sM)}\sqrt{s+1}J_{|\ell-\delta|}(p^{\perp}\rho)e^{i\ell\varphi}\\\\[4.0pt]
i\epsilon_{3}\epsilon_{|\ell-\delta|}\sqrt{E_{p}-sM)}\sqrt{s-1}J_{|\ell-\delta|+\epsilon(\ell-\delta)}(p^{\perp}\rho)e^{i(\ell+1)\varphi}\end{pmatrix}$
(18) and $\displaystyle\zeta$
$\displaystyle=\frac{1}{\sqrt{2s}}\begin{pmatrix}\epsilon_{3}\sqrt{E_{p}+sM)}\sqrt{s-1}J_{|\ell-\delta|}(p^{\perp}\rho)e^{i\ell\varphi}\\\\[4.0pt]
i\epsilon_{|\ell-\delta|}\sqrt{E_{p}-sM}\sqrt{s+1}J_{|\ell-\delta|+\epsilon(\ell-\delta)}(p^{\perp}\rho)e^{i(\ell+1)\varphi}\end{pmatrix},$
(19)

where $J_{|\ell-\delta|}$ is the Bessel function of the first kind,
$\epsilon_{3}:=\mathrm{sign}(sp_{3})$, $\ell$ is an integer, and $s$ is the
eigenvalue of the equation $\hat{S}\psi=s\psi$. In what follows, we take
$\mathrm{sign}(sp_{3})=1$. Since $p^{2}=E_{p}^{2}-M^{2}$, we have

$(p^{\perp})^{2}=p^{2}-p_{3}^{2},\qquad
s=\pm\sqrt{1+\frac{p_{3}^{2}}{M^{2}}}\,.$ (20)

The first-order Born amplitude for the scattering of a Dirac particle induced
by spin noncommutativity is then

$\displaystyle S_{fi}^{(1)}$ $\displaystyle=-i\int
dt\,\langle\psi_{f}|H_{\mathrm{int}}|\psi_{i}\rangle$ $\displaystyle=-i\int
dt\,d^{3}x\,\psi_{f}^{\dagger}(x)H_{\mathrm{int}}\psi_{i}(x)e^{i(E_{f}-E_{i})t}e^{i(p^{\prime}_{3}-p_{3})z}$
$\displaystyle=-i(2\pi)^{2}i\delta(E_{f}-E_{i})\delta(p^{\prime}_{3}-p_{3})f_{fi}^{(1)}.$
(21)

This result emerges from the $t$ and $z$ integrations corresponding to energy
and momentum conservation in the $z$ direction. Thus, we are left with
calculation of the following integral:

$f_{fi}^{(1)}=\int\rho\,d\rho\,d\varphi\,\psi_{f}^{\dagger}(p^{\perp}\rho)H_{\mathrm{int}}\psi_{i}(p^{\perp}\rho).$
(22)

This integration is over all space, and $\psi$ is the Dirac spinor given by
Eq. (17). Furthermore, we introduce the following notation for some terms in
the spinor components:

$\displaystyle\sqrt{E_{p}+sM}$
$\displaystyle:=\surd{E_{p+}},\qquad\sqrt{s+1}:=\surd{s_{+}},$ (23)
$\displaystyle\sqrt{E_{p}-sM}$
$\displaystyle:=\surd{E_{p-}},\qquad\sqrt{s-1}:=\surd{s_{-}},$ (24)
$\displaystyle\epsilon_{\ell}$
$\displaystyle:=\epsilon_{|\ell-\delta|}=\begin{cases}1,&\ell>\delta,\\\
-1,&\ell<\delta,\end{cases}$ (25) $\displaystyle J_{\nu_{i}}$
$\displaystyle:=\begin{cases}J_{|\ell-\delta|},&i=1,\\\
J_{|\ell-\delta|-\epsilon(\ell-\delta)},&i=2,\end{cases}$ (26)

and similarly for their Hermitian adjoints $\epsilon_{\tilde{\ell}}$ and
$J_{\tilde{\nu}_{i}}$ ($\tilde{\ell}$ is the azimuthal quantum number of the
outgoing wave). With these considerations, the invariant amplitude can be
written as

$S_{fi}^{(1)}=-\frac{ie\theta\Phi}{8\pi
E_{p}s}\int_{0}^{\infty}\int_{0}^{2\pi}\frac{1}{\rho^{2}}\,d\rho\,d\varphi\mathbf{v}^{\dagger}\begin{pmatrix}0&e^{-i\ell\varphi}z^{+}_{ij}&0&0\\\\[2.0pt]
e^{i\ell\varphi}z^{-}_{ij}&0&0&0\\\\[2.0pt]
0&0&0&e^{-i\ell\varphi}z^{+}_{ij}\\\\[2.0pt]
0&0&e^{i\ell\varphi}z^{-}_{ij}&0\end{pmatrix}\mathbf{u},$ (27)

where

$\displaystyle\mathbf{u}$
$\displaystyle=\begin{pmatrix}\surd{E_{p+}}\surd{s_{+}}J_{\nu_{1}}e^{i\ell\varphi}\\\\[3.0pt]
i\epsilon_{\ell}\surd{E_{p-}}\surd{s_{-}}J_{\nu_{2}}e^{i(\ell+1)\varphi}\\\\[3.0pt]
\surd{E_{p+}}\surd{s_{-}}J_{\nu_{1}}e^{i\ell\varphi}\\\\[3.0pt]
i\epsilon_{\ell}\surd{E_{p-}}\surd{s_{+}}J_{\nu_{2}}e^{i(\ell+1)\varphi}\end{pmatrix}$
(28) [which, up to constants, is the spinor in Eq. (17)] and
$\displaystyle\mathbf{v}$
$\displaystyle=\begin{pmatrix}\surd{E_{p+}}\surd{s_{+}}J_{\tilde{\nu}_{1}}e^{i\tilde{\ell}\tilde{\varphi}}\\\\[3.0pt]
i\epsilon_{\tilde{\ell}}\surd{E_{p-}}\surd{s_{-}}J_{\tilde{\nu}_{2}}e^{i(\tilde{\ell}+1)\tilde{\varphi}}\\\\[3.0pt]
\surd{E_{p+}}\surd{s_{-}}J_{\tilde{\nu}_{1}}e^{i\tilde{\ell}\tilde{\varphi}}\\\\[3.0pt]
i\epsilon_{\tilde{\ell}}\surd{E_{p-}}\surd{s_{+}}J_{\tilde{\nu}_{2}}e^{i(\tilde{\ell}+1)\tilde{\varphi}}\end{pmatrix},$
(29)

and $\dagger$ indicates the Hermitian conjugate. Here,
$\tilde{\varphi}=\varphi-\Omega$, where $\Omega$ is the scattering angle (see
Fig. 1).

Figure 1: Geometry of the Aharonov–Bohm effect. A magnetic field $\vec{B}$ is
confined in a straight solenoid along the $z$ axis, which contains a magnetic
flux $\Phi$. The cross section of the solenoid is shown here as the black
circle and there is no magnetic field outside it. Relativistic charged
particles with energy $E_{p}$ and mass $M$ are incident from the negative $x$
direction. They are scattered by the solenoid, but cannot penetrate it.

Equation (27) involves spatial derivatives given by the $z$ operators defined
in Eq. (16). After multiplying all the matrices, we get

$\displaystyle S_{fi}^{(1)}=\frac{e\theta\Phi}{4\pi
E_{p}s}\mathcal{W}\int_{0}^{\infty}\int_{0}^{2\pi}d\rho\,d\varphi$
$\displaystyle\left\\{\epsilon_{\ell}\frac{1}{\rho^{2}}[(\ell+1)J_{\tilde{\nu}_{1}}J_{\nu_{2}}+J_{\tilde{\nu}_{1}}J_{\nu_{2}}^{\prime}p^{\perp}\rho]e^{-i\tilde{\ell}\tilde{\varphi}+i\ell\varphi}\right.$
$\displaystyle\hskip
6.0pt\left.-\epsilon_{\tilde{\ell}}\frac{1}{\rho^{2}}[\ell
J_{\tilde{\nu}_{2}}J_{\nu_{1}}-J_{\tilde{\nu}_{2}}J_{\nu_{1}}^{\prime}p^{\perp}\rho]e^{-i(\tilde{\ell}+1)\tilde{\varphi}+i(\ell+1)\varphi}\right\\},$
(30)

where $\mathcal{W}=\surd{E_{p+}}\surd{E_{p-}}\surd{s_{+}}\surd{s_{-}}$. The
above expression can be cast as

$\displaystyle S_{fi}^{(1)}=\frac{e\theta\Phi}{4\pi
E_{p}s}\mathcal{W}\int_{0}^{\infty}\int_{0}^{2\pi}d\rho\,d\varphi$
$\displaystyle\left\\{\left[\frac{\epsilon_{\ell}}{\rho^{2}}(\ell+1)J_{\tilde{\nu}_{1}}J_{\nu_{2}}+\frac{\epsilon_{\ell}}{\rho}J_{\tilde{\nu}_{1}}J_{\nu_{2}}^{\prime}p^{\perp}\right]e^{i(\ell-\tilde{\ell})\varphi+i\tilde{\ell}\Omega}\right.$
$\displaystyle\hskip
6.0pt\left.+\left(-\frac{\epsilon_{\tilde{\ell}}}{\rho^{2}}\ell
J_{\tilde{\nu}_{2}}J_{\nu_{1}}+\frac{\epsilon_{\tilde{\ell}}}{\rho}J_{\tilde{\nu}_{2}}J_{\nu_{1}}^{\prime}p^{\perp}\right)e^{i(\ell-\tilde{\ell})\varphi+i(\tilde{\ell}+1)\Omega}\right\\},$
(31)

where the prime denotes differentiation with respect to the argument of the
Bessel functions of the first kind, which are given explicitly in Eqs. (18)
and (19). We can write $\tilde{\varphi}$ in terms of the scattering angle
$\Omega$, and in this way we can easily perform the $\varphi$ integration,
yielding

$\displaystyle
S_{fi}^{(1)}=\frac{e\theta\Phi}{2E_{p}s}\mathcal{W}\lim_{\lambda\rightarrow
1}\int_{0}^{\infty}d\rho$
$\displaystyle\left\\{\left[\frac{\epsilon_{\ell}}{\rho^{2\lambda}}(\ell+1)J_{\nu_{1}}J_{\nu_{2}}+\frac{\epsilon_{\ell}}{2\rho^{2\lambda-1}}(J_{\nu_{1}}J_{{\nu_{2}}-1}-J_{\nu_{1}}J_{{\nu_{2}}+1})p^{\perp}\right]e^{i\ell\Omega}\right.$
$\displaystyle\hskip
6.0pt\left.+\left[\frac{-\epsilon_{\ell}}{\rho^{2\lambda}}(\ell+1)J_{\nu_{1}}J_{\nu_{2}}+\frac{\epsilon_{\ell}}{2\rho^{2\lambda-1}}(J_{\nu_{2}}J_{{\nu_{1}}-1}-J_{\nu_{2}}J_{{\nu_{1}}+1})p^{\perp}\right]e^{i(\ell+1)\Omega}\right\\},$
(32)

where we have introduced a parameter $\lambda$ so that, after all the
integrations have been performed, we can investigate whether its physical
value $\lambda=1$ can be taken. The $\rho$ integrals can be performed
GradshteynRyzhik  using the analytical continuation of the formula

$\int_{0}^{\infty}\frac{J_{\nu}(kr)J_{\mu}(kr)}{r^{\sigma}}dr=\frac{k^{\sigma-1}\Gamma(\sigma)\Gamma(A)}{2^{\sigma}\Gamma(B)\Gamma(C)\Gamma(D)},$
(33)

where the arguments in the gamma functions and the calculation method are
shown explicitly in Appendix A.

After performing all the integrations in Eq. (II), we obtain for the invariant
amplitude

$S_{fi}=\frac{e\theta\Phi
p^{\perp}}{8E_{p}}\frac{\mathcal{W}}{s}\sum_{\ell}\frac{\epsilon_{\ell}[(2\ell-\delta+2)e^{i\ell\Omega}-(\ell+1)e^{i(\ell+1)\Omega}]}{(\ell-\delta)(\ell-\delta+1)}.$
(34)

After some manipulations, we find from Eq. (20) that

$\frac{\mathcal{W}}{s}=\frac{p^{\perp}p_{3}}{\sqrt{M^{2}+p_{3}^{2}}},$ (35)

where $p^{\perp}=|\vec{p}^{\perp}|$ and the $z$ component of momentum are
related to the relativistic energy by
$E_{p}^{2}=(p^{\perp})^{2}+p_{3}^{2}+M^{2}$.

##  III Conclusions

In this paper, we have studied spin noncommutativity for a relativistic system
through deformation of the spacetime coordinates $x^{\mu}$ into a nonstandard
Heisenberg algebra given by Eq. (2).

In the nonrelativistic situation, the AB scattering amplitude for spin-$1/2$
particles subject to this kind of noncommutativity was calculated in Ref.
PhysLett.B680.384, . Generalizing that study to the case of spin-$s$
relativistic particles, we have calculated in $(3+1)$ dimensions the
scattering amplitude in the AB symmetric gauge.

For canonical noncommutativity, the AB problem for a solenoid of finite radius
was considered in Ref. PhysRev.D66.045018, . In that regime, the gauge
potential was expanded in powers of the noncommutativity parameter and the
magnetic flux $\Phi$. There, the first order in $\theta$ appeared multiplied
by $\Phi^{2}$, and thus, since we are considering here a small magnetic flux,
this term is outside the scope of the present model.

We have restricted attention to the potential as written in Eq. (10). In our
calculations, because of symmetry considerations, the energy and $p_{3}$ are
conserved.

From Eq. (34), we see that the scattering amplitude depends explicitly on
$\delta$, the noninteger part of $\alpha$, the term involving the flux. This
is quite different from the case of canonical noncommutativity in $(2+1)$
dimensions, where $\delta$ is absorbed into $\alpha$. PhysRev.D76.085008

The canonical NC algebra in Eq. (1) has generated many questions regarding
violations of unitarity and causality. This is primarily a consequence of the
NC parameter $\theta$ being a constant matrix. However, the present model
considering spin noncommutativity has the advantage over the canonical one in
that it conserves Lorentz symmetry, since the algebra in Eq. (2) is written in
a covariant way. In addition, there is no UV/IR mixing; i.e., the limit
$\theta\rightarrow 0$ is smooth.

The study of relativistic or even nonrelativistic quantum mechanics with NC
coordinates allows the examination of phenomena that occur at very small
scales and the exploration of their physical consequences in a simple setting.
However, it should be noted that physical effects related to quantum gravity
will appear only in very high-energy processes.

###### Acknowledgements.

I would like to thank Professor M. Gomes for useful discussions and
suggestions.

##  Appendix A Integrals of Bessel functions in Eq. (II)

The wave contributions for $\ell\leq 0$ and $\ell>0$, given by the integrals
of the double products of Bessel functions in Eq. (II), can be calculated
using the following result:

$\int_{0}^{\infty}\frac{J_{\nu}(kr)J_{\mu}(kr)}{r^{\sigma}}\,dr=\frac{k^{\sigma-1}\Gamma(\sigma)\Gamma(A)}{2^{\sigma}\Gamma(B)\Gamma(C)\Gamma(D)},$
(36)

where

$\displaystyle 2A$ $\displaystyle=\mu+\nu-\sigma+1,$ (37) $\displaystyle 2B$
$\displaystyle=-\nu+\mu+\sigma+1,$ (38) $\displaystyle 2C$
$\displaystyle=\mu+\nu+\sigma+1,$ (39) $\displaystyle 2D$
$\displaystyle=\nu-\mu+\sigma+1$ (40)

is formally valid provided that the inequalities

$\mathrm{Re}(\mu+\nu+1)>\mathrm{Re}(\sigma)>0$ (41)

are satisfied.

###  A.1 The first integral in Eq. (II)

The contribution for $\ell\leq 0$ leads to the following integral:

$\lim_{\lambda\rightarrow
1}\int_{0}^{\infty}dr\,\frac{e^{i\ell\Omega}(\ell+1)}{r^{2\lambda}}J_{\alpha-\ell}(kr)J_{\alpha-\ell-1}(kr)=\frac{e^{i\ell\Omega}(\ell+1)}{2}\frac{(\frac{1}{2}k)^{\sigma-1}\Gamma(A)}{\Gamma(B)\Gamma(C)\Gamma(D)},$
(42)

where $\sigma=2\lambda$ and

$\displaystyle 2A$ $\displaystyle=\mu+\nu-\sigma+1=2\alpha-2\ell-2\lambda,$
(43) $\displaystyle 2B$ $\displaystyle=-\mu+\nu+\sigma+1=2\lambda,$ (44)
$\displaystyle 2C$ $\displaystyle=\mu+\nu+\sigma+1=2\lambda+2\alpha-2\ell,$
(45) $\displaystyle 2D$ $\displaystyle=\mu-\nu+\sigma+1=2\lambda+2.$ (46)

The condition (41) (with $\lambda\rightarrow 1$) implies

$\alpha-\ell+\alpha-\ell-1+1>2,$ (47)

i.e.,

$\alpha>1+\ell,$ (48)

so that for $\ell<0$ (and excluding the case $\ell=0$), we have $0<\alpha<1$.
This kind of divergence can be eliminated by not taking $\lambda\rightarrow 1$
directly. Instead, we first perform the integral and then take the limit. The
integral in Eq. (42) then becomes

$\lim_{\lambda\rightarrow
1}\int_{0}^{\infty}dr\,\frac{e^{i\ell\Omega}(\ell+1)}{r^{2\lambda}}J_{\alpha-\ell}(kr)J_{\alpha-\ell-1}(kr)=\frac{e^{i\ell\Omega}(\ell+1)}{4}\frac{k\Gamma(\alpha-\ell-1)}{\Gamma(\alpha-\ell+1)\Gamma(2)},$
(49)

where
$\Gamma(\alpha-\ell+1)=(\alpha-\ell)(\alpha-\ell-1)\Gamma(\alpha-\ell-1)$. The
other integrals can be calculated following the same approach.

##  Appendix B The $z_{ij}$ operator in Eq. (16)

To obtain the expression for the $z_{ij}$ operator in Eq. (16), we have to use
expressions for the Pauli matrices $\sigma^{i}$ and remember that the $x_{i}$
($i=1,2$) are in cylindrical coordinates. We have

$\displaystyle H_{\mathrm{int}}=\frac{e\theta}{2}\frac{\Phi}{\pi}$
$\displaystyle\left\\{\frac{2x_{1}x_{2}}{\rho^{4}}\begin{pmatrix}0&-1&0&0\\\
1&0&0&0\\\ 0&0&0&-1\\\
0&0&1&0\end{pmatrix}\partial_{2}+\left(\frac{1}{\rho^{2}}-\frac{2x_{1}^{2}}{\rho^{4}}\right)\begin{pmatrix}0&i&0&0\\\
i&0&0&0\\\ 0&0&0&i\\\ 0&0&i&0\end{pmatrix}\partial_{2}\right.$
$\displaystyle\hskip
6.0pt\left.+\left(\frac{1}{\rho^{2}}-\frac{2x_{2}^{2}}{\rho^{4}}\right)\begin{pmatrix}0&-1&0&0\\\
1&0&0&0\\\ 0&0&0&-1\\\
0&0&1&0\end{pmatrix}\partial_{1}+\frac{2x_{1}x_{2}}{\rho^{4}}\begin{pmatrix}0&i&0&0\\\
i&0&0&0\\\ 0&0&0&i\\\ 0&0&i&0\end{pmatrix}\partial_{1}\right\\},$ (50)

where $\rho^{2}=x_{1}^{2}+x_{2}^{2}.$ The nonzero matrix elements are

$\displaystyle a_{12}=a_{34}$
$\displaystyle=[-2x_{1}x_{2}+i(x_{2}^{2}+x_{1}^{2})]\partial_{2}+[(x_{2}^{2}-x_{1}^{2})+2ix_{1}x_{2}]\partial_{1}$
$\displaystyle=[-x_{1}x_{2}+ix_{2}^{2}-x_{1}x_{2}-ix_{1}^{2}]\partial_{2}+[ix_{1}x_{2}+x_{2}^{2}+ix_{1}x_{2}-x_{1}^{2}]\partial_{1}$
$\displaystyle=[-x_{2}(x_{1}-ix_{2})-ix_{1}(x_{1}-ix_{2})]\partial_{2}+[ix_{2}(x_{1}-ix_{2})-x_{1}(x_{1}-ix_{2})]\partial_{1}$
$\displaystyle=\rho
e^{-i\varphi}[(-x_{2}-ix_{1})\partial_{2}+(ix_{2}-x_{1})\partial_{1}]$
$\displaystyle=\rho
e^{-i\varphi}[-i\epsilon_{ij}x_{i}\partial_{j}-\delta_{ij}x_{i}\partial_{j}]$
$\displaystyle=-\rho
e^{-i\varphi}[i\epsilon_{ij}+\delta_{ij}]x_{i}\partial_{j}=-\rho
e^{-i\varphi}z_{ij}^{(+)}$ (51) and $\displaystyle a_{21}=a_{43}$
$\displaystyle=[2x_{1}x_{2}+i(x_{2}^{2}-x_{1}^{2})]\partial_{2}+[(x_{1}^{2}-x_{2}^{2})+2ix_{1}x_{2}]\partial_{1}$
$\displaystyle=[x_{1}x_{2}+ix_{2}^{2}+x_{1}x_{2}-ix_{1}^{2}]\partial_{2}+[ix_{1}x_{2}+x_{1}^{2}+ix_{1}x_{2}-x_{2}^{2}]\partial_{1}$
$\displaystyle=[x_{2}(x_{1}+ix_{2})-ix_{1}(x_{1}+ix_{2})]\partial_{2}+[x_{1}(x_{1}+ix_{2})+x_{2}(x_{1}+ix_{2})]\partial_{1}$
$\displaystyle=\rho
e^{i\varphi}[(x_{2}-ix_{1})\partial_{2}+(x_{1}+ix_{2})\partial_{1}]$
$\displaystyle=-\rho
e^{i\varphi}[i\epsilon_{ij}x_{i}\partial_{j}-\delta_{ij}x_{i}\partial_{j}]=-\rho
e^{i\varphi}z_{ij}^{(-)},$ (52)

where $z_{ij}^{(\pm)}=(i\epsilon_{ij}\pm\delta_{ij})x_{i}\partial_{j}$.

##  Appendix C Action of the $z_{ij}^{(\pm)}$ operator on a spinor

It is interesting now to see how the $z_{ij}^{(\pm)}$ operator acts on a
function that depends on $\varphi$ and $\rho$. In the context of this work,
the following are some appropriate cases:

$\displaystyle\partial_{j}(e^{i\ell\varphi}J_{|\ell-\delta|})$
$\displaystyle=\frac{-i\ell\epsilon_{jm}x_{m}}{\rho^{2}}e^{i\ell\varphi}J_{|\ell-\delta|}+e^{i\ell\varphi}J_{|\ell-\delta|}^{\prime}p^{\perp}\frac{x_{j}}{\rho},$
(53) $\displaystyle\partial_{j}\rho$ $\displaystyle=\frac{x_{j}}{\rho},$ (54)
$\displaystyle\partial_{j}\varphi$
$\displaystyle=\frac{-\epsilon_{jk}x_{k}}{\rho^{2}},$ (55)
$\displaystyle\partial_{j}(e^{i(\ell+1)\varphi}J_{|\ell-\delta|+\epsilon_{\ell}})$
$\displaystyle=\left[\frac{-i(\ell+1)\epsilon_{jm}x_{m}}{\rho^{2}}J_{|\ell-\delta|+\epsilon_{\ell}}+J_{|\ell-\delta|+\epsilon_{\ell}}^{\prime}p^{\perp}\frac{x_{j}}{\rho}\right]e^{i(\ell+1)\varphi}.$
(56)

The product of the two last matrices in Eq. (27) yields the following nonzero
matrix elements:

$\displaystyle a_{11}={}$ $\displaystyle
i\epsilon_{\ell}\surd{E_{p-}}\surd{s_{-}}\frac{e^{-i\varphi}}{\rho^{2}}(i\epsilon_{ij}+\delta_{ij})x_{i}\partial_{j}[e^{i(\ell+1)\varphi}J_{\nu_{2}}(p^{\perp}\rho)]$
$\displaystyle={}$ $\displaystyle
i\epsilon_{\ell}\surd{E_{p-}}\surd{s_{-}}\frac{e^{-i\varphi}}{\rho^{2}}(i\epsilon_{ij}+\delta_{ij})\left[\frac{-i(\ell+1)}{\rho^{2}}\epsilon_{jm}x_{i}x_{m}J_{\nu_{2}}+J_{\nu_{2}}^{\prime}p^{\perp}\frac{x_{i}x_{j}}{\rho}\right]e^{i(\ell+1)\varphi}$
$\displaystyle={}$ $\displaystyle
i\epsilon_{\ell}\surd{E_{p-}}\surd{s_{-}}\frac{1}{\rho^{2}}[(\ell+1)J_{\nu_{2}}+J_{\nu_{2}}^{\prime}p^{\perp}\rho]e^{i\ell\varphi},$
(57) $\displaystyle a_{21}={}$
$\displaystyle\surd{E_{p+}}\surd{s_{+}}\frac{e^{i\varphi}}{\rho^{2}}(i\epsilon_{ij}-\delta_{ij})x_{i}\partial_{j}[e^{i\ell\varphi}J_{\nu_{1}}(p^{\perp}\rho)]$
$\displaystyle={}$
$\displaystyle\surd{E_{p+}}\surd{s_{+}}\frac{e^{i\varphi}}{\rho^{2}}(i\epsilon_{ij}-\delta_{ij})\left[\frac{-i\ell\epsilon_{jm}}{\rho^{2}}x_{i}x_{m}J_{\nu_{1}}+J_{\nu_{1}}^{\prime}p^{\perp}\frac{x_{i}x_{j}}{\rho}\right]e^{i\ell\varphi}$
$\displaystyle={}$
$\displaystyle\surd{E_{p+}}\surd{s_{+}}\frac{1}{\rho^{2}}[\ell
J_{\nu_{1}}-J_{\nu_{1}}^{\prime}p^{\perp}\rho]e^{i(\ell+1)\varphi},$ (58)
$\displaystyle a_{31}={}$ $\displaystyle
i\surd{E_{p-}}\surd{s_{+}}\frac{e^{-i\varphi}}{\rho^{2}}(i\epsilon_{ij}+\delta_{ij})\left[\frac{-i(\ell+1)}{\rho^{2}}\epsilon_{jm}x_{i}x_{m}J_{\nu_{2}}+J_{\nu_{2}}^{\prime}p^{\perp}\frac{x_{i}x_{j}}{\rho}\right]e^{i(\ell+1)\varphi}$
$\displaystyle={}$ $\displaystyle
i\frac{\surd{E_{p-}}\surd{s_{+}}}{\rho^{2}}[(\ell+1)J_{\nu_{2}}+J_{\nu_{2}}^{\prime}p^{\perp}\rho]e^{i(\ell+1)\varphi},$
(59) $\displaystyle a_{41}={}$
$\displaystyle\surd{E_{p+}}\surd{s_{-}}\frac{e^{i\varphi}}{\rho^{2}}(i\epsilon_{ij}-\delta_{ij})\left[\frac{-i\ell}{\rho^{2}}\epsilon_{jm}x_{m}x_{i}J_{\nu_{1}}+J_{\nu_{1}}^{\prime}p^{\perp}\frac{x_{i}x_{j}}{\rho}\right]e^{i\ell\varphi}$
(60) $\displaystyle={}$
$\displaystyle\frac{\surd{E_{p+}}\surd{s_{-}}}{\rho^{2}}[\ell
J_{\nu_{1}}-J_{\nu_{1}}^{\prime}p^{\perp}\rho]e^{i(\ell+1)\varphi},$ (61)

where $\epsilon_{ij}\epsilon_{jm}x_{i}x_{j}=\rho^{2}$ and
$\epsilon_{jm}\delta_{ij}x_{i}x_{m}=\epsilon_{ij}x_{i}x_{j}=0$.

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