(and the polluted zone), the human was only providing 30% of the power
required to move the wheel of the bike with the other 70% being provided by
the motor. As clearly seen from Figure 6.7, when $m^{*}$ drops to 0.3, the
breathing rate decreased to approximately 30 litres/minute.

####  6.2.5 Conclusion on use case

This use case built on the use case from Section 5.2. The objective of both
use cases was to propose a method by which the amount of pollutants that a
cyclist inhaled could be reduced while cycling through a polluted area. The
method implementation was different from the open loop case, in that it
involved implementing closed loop control. The closed loop control relied
heavily on the model derived in Chapter 4. The objective of the control
circuit was to ensure that the actually observed value $\bar{m}$ tracked the
target value of $m^{*}$. It was shown (see Figure 6.3) that the control
circuit was able to track the target value $m^{*}$ with an error within $\pm$
10%, 90% of the time. It was not expected to be able to achieve perfect
tracking between the actual value $\bar{m}$ and the target value $m^{*}$ due
to the nature of the system.

The effect these changes in $m^{*}$ had on the cyclist’s breathing rate was
monitored using a spirometer. It was found that the cyclist’s breathing rate
approximately halved when the target value $m^{*}$ was reduced from 0.9 to
0.3. This proves the hypothesis that was being tested which was to show that
the cyclist’s breathing rate could be reduced by decreasing the value of
$m^{*}$. In a polluted area, if the cyclist’s breathing rate is reduced, the
amount of air pollutants that they inhale are also reduced. As such, a new
method has been presented that reduces the amount of pollutants that a cyclist
inhales while cycling through a polluted area.

Many of the extensions to the use case that were listen in Section 5.2.5 are
also applicable here, particularly the next steps to expand the simulation by
carrying out a real world cycling experiment. Significance testing should also
be carried out based on the reduction in breathing rate and the associated
reduction in inhaled air pollutants. Further refinement of the specific
parameters in the control circuit could also be carried out including 1) the
sampling period 2) the gain parameter and 3) the way in which $m^{*}$ is
reduced and increased when entering the transient zone and leaving the
polluted zone.

Using the spirometer to record the breathing rate and the finger pulse
oximeter to record the heart rate also helped to validate that heart rate was
a good indicator of breathing rate. As seen from Figure 6.6, both signals
approximately track each other. This meant that for real life non-stationary
cycling simulations, measuring the cyclist’s heart rate using an activity
tracker would be acceptable to validate that the cyclist’s breathing rate had
been reduced. This was done for the open loop case in Section 5.2.4. The
transient between the breathing rate and heart rate should be studied further
and incorporated into the methodology to determine the size of the transient
area in the simulation.

This use case will be presented at the Eighteenth Yale Workshop on Adaptive
and Learning Systems 333https://www.eng.yale.edu/css/ at Yale University, USA
in June 2017.

## Chapter 7Conclusions

This project had one main objective which was to demonstrate different
services that electric bikes can provide in smart cities to help and protect
people. In order to demonstrate these services, it was first required to
define and implement the concept of a smart electric bike. A smart electric
bike was defined as an electric bike that is situation aware and can use this
awareness to respond to its surroundings to deliver a service to help and
protect people.

A number of design criteria were stated in Section 3.1. A smart electric bike
which met those design criteria was designed and fully instrumented. The
hardware components of the smart electric bike were discussed in detail in
Section 3.2 and the software components were discussed in Section 3.3. A key
design decision was to connect the electric bike and the smartphone by
Bluetooth in order to widen the range of services that the smart electric bike
could offer. This meant that the electric bike could provide services using
any data stream that was available on the internet. A number of proposed
improvements to the system are discussed in Section 3.4.

When the system was fully instrumented, it was then possible to show how the
electric bike could provide different services in cities to help and protect
people. Chapter 4, Chapter 5 and Chapter 6 focused on these services. Section
5.2 proposed a method which aimed to reduce the amount of pollutants that a
cyclist inhaled when they were cycling through a polluted area. This method
was based on the concept that in order to reduce the amount of pollutants that
a cyclist inhaled that their breathing rate should be reduced. This method was
tested and validated in Section 5.2.4.

The open loop use case presented above was demonstrated as a proof of concept
for Dublin City Council and the Fraunhofer-Morgenstadt Initiative as part of a
smart transport solutions demonstration in Trinity College Dublin in mid-March
2017. When this proof of concept was established, it was decided to expand on
this use case by implementing closed loop control. In order to implement
closed loop control it was necessary to have a reliable model of the system
behaviour, this was developed in Chapter 4. Section 6.2 then relied heavily on
this model to implement closed loop control.

For the closed loop use case in Section 6.2, the variable $m$ introduced in
Chapter 4 was made use of to manage the proportion of power that the cyclist
and the motor were providing to move the wheel of the bike. It was
demonstrated that it was possible to regulate the observed value $\bar{m}$ so
that it tracked a target value $m^{*}$. The error between the target and the
actual value of $m$ was maintained within $\pm$ 10%, 90% of the time. It is
believed that this could be improved by further refining the control
parameters including the gain $\gamma$, the sampling period $T_{s}$ and the
way in which the target value $m^{*}$ changes when entering and leaving a
polluted area.

It was validated that the cyclist’s breathing rate was reduced in different
ways for both the open loop and the closed loop cases. In the closed loop
case, the cyclist was equipped with spirometry equipment to measure the
volumetric rate at which they were breathing in air. They were also equipped
with a finger pulse oximeter which monitored their heart rate. It was found
that the user’s breathing rate more than doubled from 25 litres/minute to 65
litres/minute when they were cycling from rest in a non-polluted area
($m^{*}=0.9$). By reducing the value of $m^{*}$ to 0.3, it was found that this
reduced the cyclist’s breathing rate to approximately 30 litres/minute.

It was also found that for the steady state case, there was a strong
correlation between the breathing rate and heart rate signals which agreed
with the review of literature discussed in Section 2.6. This fact meant that
(to a first approximation) it was possible to use a measurement of the heart
rate to validate that the cyclist’s breathing rate had been reduced. This was
made use of for validation in the open loop case where the cyclist wore an
activity tracker which measured their heart rate. This meant that for non-
stationary cycling tests, the cyclist would not need to be equipped with
spirometry equipment. The transient length in the breathing rate signal and
the heart rate signal seem to be different and should be studied further. If
they are different, this would impact the size of the required transient zone.

Many extensions of this use case are possible. This includes changes to the
model from Chapter 4 to generalise it further. The model should be expanded to
include different cycling environments particularly including roads with
different positive and negative gradients. The use case could be demonstrated
further by carrying out a real life cycling experiment where a user would
cycle on a specified path through an area with a specific part of the area
being designated as a polluted zone. The system could then use location data
available on the smartphone to update the value of $m^{*}$ while in that area.
This use case will be presented at the Eighteenth Yale Workshop on Adaptive
and Learning Systems at Yale University in the USA in June 2017.

It is possible to demonstrate many other use cases using the smart electric
bike. An example would be a service to manage the way in which energy is
delivered to the cyclist over a cycling trip. This would involve the system
becoming aware of the final destination of the cyclist, and then using that
knowledge to optimally deliver the energy in the battery to the cyclist over
the course of the journey. This could involve saving energy in the battery for
times when the cyclist encounters large hills on their route and therefore
needs extra assistance. This use case could be particularly useful for
vulnerable groups of road users like older people or people who have physical
disabilities.

Another interesting use case would be to examine the potential of the smart
electric bike for smart routing applications. In this use case, the system
would become aware of the cyclist’s final destination on their journey and
would then use smart routing to recommend the optimal route for the cyclist to
travel on to achieve some objective. This objective might be to go on a route
that minimises the cyclist’s inhalation of pollutants, or that has the best
cycling paths or the least number of cars.

Another use case would be to make use of some of the data from the activity
tracker (discussed in Section 5.2). Activity trackers like the Microsoft Band
estimate the amount of metabolic calories that the user is burning off. It
would be possible to implement a closed loop control circuit whereby a cyclist
could set a target rate that they wish to burn calories at over a journey and
the control circuit could regulate the amount of power that the motor is
delivering to the human in order to ensure that they stay on track with their
target rate of calories burned. It would also be valuable to investigate the
correlation between this estimation of metabolic calories burned and the human
power input at the pedals. If the activity tracker calories burned measurement
and the human power input measurement tracked each other well, it might be
possible to substitute the THUN X-CELL RT sensor for an activity tracker for
some use cases which would cut down on costs.

The use case regarding reducing a cyclist’s inhalation of pollutants generated
particular interest which was why it was focused on so much in Chapters 5 and
6. In general, there are a large number of potential services that electric
bikes can provide in our cities to help and protect people. They can also help
to solve or lessen a number of problems in cities like congestion due to cars,
parking problems and to reduce the concentration of air pollutants in cities
which would have a direct impact on improving human health.

## References

  * [1] James C Martin et al.  “Validation of a mathematical model for road cycling.pdf”  In _Journal of applied biomechanics_ 14, 1998, pp. 276–291  URL: http://www.academia.edu/download/39236393/43.%7B%5C_%7DMartin%7B%5C_%7Det%7B%5C_%7Dal%7B%5C_%7DJ%7B%5C_%7DAppl%7B%5C_%7DBiomech%7B%5C_%7D1998.pdf
  * [2] Ankang Le  “Model Predictive Control for Sensor-Based Group Training Optimization in Professional Cycling”  In _Proceedings of the European Control Conference 2009_ , 2009, pp. 1305–1310
  * [3] Tim Olds  “Olds (1998) - The mathematics of breaking away and chasing in cycling”  In _European Journal of Applied Physiology and Occupational Physiology_ 77.6, 1998, pp. 492–497
  * [4] W.C. Morchin  “Energy management in hybrid electric vehicles”  In _17th DASC. AIAA/IEEE/SAE. Digital Avionics Systems Conference. Proceedings (Cat. No.98CH36267)_ 2.253 IEEE, 1998, pp. 1–6  DOI: 10.1109/DASC.1998.739880
  * [5] Marilena Kampa and Elias Castanas  “Human health effects of air pollution”  In _Environmental Pollution_ 151.2, 2008, pp. 362–367  DOI: 10.1016/j.envpol.2007.06.012
  * [6] A Seaton, W MacNee, K Donaldson and D Godden  “Particulate air pollution and acute health effects.”  In _Lancet_ 345.8943, 1995, pp. 176–178  DOI: 10.1016/S0140-6736(95)90173-6
  * [7] Hong Chen et al.  “Articles Living near major roads and the incidence of dementia, Parkinson’s disease, and multiple sclerosis: a population-based cohort study”  In _Lancet_ 389, 2017, pp. 718–726  DOI: 10.1016/S0140-6736(16)32399-6
  * [8] Norbert Englert  “Fine particles and human health - A review of epidemiological studies”  In _Toxicology Letters_ 149.1-3, 2004, pp. 235–242  DOI: 10.1016/j.toxlet.2003.12.035
  * [9] Bert Brunekreef and Stephen T. Holgate  “Air pollution and health”  In _Lancet_ 360.9341, 2002, pp. 1233–1242  DOI: 10.1016/S0140-6736(02)11274-8
  * [10] Jeroen Johan Hartog, Hanna Boogaard, Hans Nijland and Gerard Hoek  “Do the health benefits of cycling outweigh the risks?”  In _Environmental Health Perspectives_ 118.8, 2010, pp. 1109–1116  DOI: 10.1289/ehp.0901747
  * [11] Luc Int Panis et al.  “Exposure to particulate matter in traffic: A comparison of cyclists and car passengers”  In _Atmospheric Environment_ 44.19, 2010, pp. 2263–2270  DOI: 10.1016/j.atmosenv.2010.04.028
  * [12] C C Daigle et al.  “Ultrafine particle deposition in humans during rest and exercise”  In _Inhalation Toxicology_ 15.6, 2003, pp. 539–552  DOI: 10.1080/08958370390205065
  * [13] Adam Herrmann  “Situational Aware In-Car System to Protect Cyclists”  In _Masters Thesis_ , 2017
  * [14] Alexander Y. Bigazzi, Miguel A. Figliozzi, Wentai Luo and James F. Pankow  “Breath Biomarkers to Measure Uptake of Volatile Organic Compounds by Bicyclists”  In _Environmental Science and Technology_ 50.10, 2016, pp. 5357–5363  DOI: 10.1021/acs.est.6b01159
  * [15] Alexander Y. Bigazzi  “Determination of active travel speed for minimum air pollution inhalation”  In _International Journal of Sustainable Transportation_ 11.3, 2017, pp. 221–229  DOI: 10.1016/j.toxicon.2008.06.023
  * [16] Farah Bahmed, Farisa Khatoon and B Reddy  “Relation between respiratory rate and heart rate a comparative study”  In _Indian Journal of Clinical Anatomy and Physiology_ 3.4, 2016, pp. 436–439  DOI: 10.5958/2394-2126.2016.00099.2
  * [17] Jelena Nikolic-Popovic and Rafik Goubran  “Measuring heart rate, breathing rate and skin conductance during exercise”  In _2011 IEEE International Symposium on Medical Measurements and Applications, Proceedings_ , 2011  DOI: 10.1109/MeMeA.2011.5966751
  * [18] JD Schipke, G Arnold and M Pelzer  “Effect of respiration rate on short term heart rate variability.”  In _Journal of Clinical and Basic Cardiology_ 2.1, 1999, pp. 92–95
  * [19] Evgeny Vaschillo, Bronya Vaschillo and Paul Lehrer  “Heartbeat Synchronizes With Respiratory Rhythm Only Under Specific Circumstances”  In _Chest Journal_ 126.4, 2004, pp. 1385–1387
  * [20] J.C. Wallaart  “A study of the relationship between heart rate and minute breathing volume at various levels of work demonstrating the spread between individuals in a group and the implications in industry . Submitted for publication at the 1997 isrp ( international soci”  In _International Society of Respiratory Protection_ , 1997, pp. 1–17
  * [21] Ankang Le, Thomas Jaitner and Lothar Litz  “Sensor-based training optimization of a cyclist group”  In _Proceedings - 7th International Conference on Hybrid Intelligent Systems, HIS 2007_ , 2007, pp. 265–270  DOI: 10.1109/ICHIS.2007.4344062
  * [22] D R Gold et al.  “Ambient pollution and heart rate variability.”  In _Circulation_ 101, 2000, pp. 1267–1273  DOI: 10.1161/01.CIR.101.11.1267
  * [23] E W Dijkstra  “A Note on Two Probles in Connexion with Graphs”  In _Numerische Mathematik_ 1.1, 1959, pp. 269–271  DOI: 10.1007/BF01386390
  * [24] E. Crisostomi, S. Kirkland and R. Shorten  “A Google-like model of road network dynamics and its application to regulation and control”  In _International Journal of Control_ 84.3, 2011, pp. 633–651  DOI: 10.1080/00207179.2011.568005
  * [25] E. Crisostomi, S. Kirkl and R. Shorten  “Robust and risk-averse routing algorithms in road networks”  In _IFAC Proceedings Volumes (IFAC-PapersOnline)_ 18.PART 1, 2011, pp. 9512–9517  DOI: 10.3182/20110828-6-IT-1002.01147
  * [26] Ling Chen, Mingqi Lv and Gencai Chen  “A system for destination and future route prediction based on trajectory mining”  In _Pervasive and Mobile Computing_ 6.6, 2010, pp. 657–676  DOI: 10.1016/j.pmcj.2010.08.004
  * [27] Ling Chen et al.  “A personal route prediction system based on trajectory data mining”  In _Information Sciences_ 181.7, 2011, pp. 1264–1284  DOI: 10.1016/j.ins.2010.11.035


# Results from the ARIANNA high-energy neutrino detector

Christian Glaser for the ARIANNA collaboration

###### Abstract

The ARIANNA in-ice radio detector explores the detection of UHE neutrinos with
shallow detector stations on the Ross Ice Shelf and the South Pole. Here, we
present recent results that lay the foundation for future large-scale
experiments. We show a limit on the UHE neutrino flux derived from ARIANNA
data, measurements of the more abundant air showers, results from in-situ
measurement campaigns, a study of a potential background from internal
reflection layers, and give an outlook of future detector improvements.

##  1 Introduction

In-ice radio detection is a promising technique to measure the low expected
flux of ultra-high-energy neutrinos
($E_{\nu}>${10}^{17}\text{\,}\mathrm{e}\mathrm{V}$$ [1]. It was explored in
the smaller test bed arrays ARA and ARIANNA that demonstrated the technique’s
feasibility using two complementary detector station designs. Currently, a
larger array of autonomous radio detector stations is being constructed in
Greenland (RNO-G) [2] with about an order of magnitude larger effective volume
to neutrinos than previous experiments, which might allow the detection of the
first UHE neutrino. At the same time, a much larger in-ice radio array is
being planned as part of IceCube-Gen2 [3]. Here, we report on how ARIANNA
advanced the field of UHE neutrino detection and is paving the way for
IceCube-Gen2.

##  2 Versatile + Autonomous Hardware Design

The ARIANNA concept is based around compact shallow detector stations, i.e.,
antennas are deployed just underneath the snow surface with a small spatial
extend of $\mathcal{O}$($10\text{\,}\mathrm{m}$). See Fig. 1 for a sketch of
the station layout. The ARIANNA hardware evolved into a versatile Plug’n’Play
system that works at essentially any location and has been successfully used
on the Ross Ice Shelf [4], the South Pole [5] and at Mt. Melbourne in
Antarctica [6]. All electronics and batteries are housed in a pelican case
with thermal insulation with external ports for the antennas, communication,
and power inputs from solar and wind power. This simplifies deployment, as
during field installation, only the external ports need to be connected. The
station starts up automatically and can be commissioned remotely.

The ARIANNA hardware is based on the SST chip design [7]. It currently
supports up to 8 input channels on a single board. The input is sampled at
$1\text{\,}\mathrm{G}\mathrm{S}\mathrm{P}\mathrm{S}$ or
$2\text{\,}\mathrm{G}\mathrm{S}\mathrm{P}\mathrm{S}$ and continuously stored
in a switched capacitor array. The SST chip has a precise time synchronization
between samples and across channels of less than
$5\text{\,}\mathrm{p}\mathrm{s}$, which allows for a precise reconstruction of
the signal direction despite the small spatial extent of an ARIANNA station.
Following a trigger, 256 samples are digitized with 12 bit ADCs and read into
an FPGA and, after that, into an Mbed microprocessor to calculate a second
trigger stage and data storage.

The system has shown to work reliably over several years in Antarctica with
only $5\text{\,}\mathrm{W}$ power consumption. Communication is established
via the IRIDIUM satellite network or long-range WiFi if available. The station
is powered by solar panels and experimentally through wind turbines to provide
uptime during the polar winter. A custom wind turbine – developed and built at
Uppsala University – was successfully tested at Moore’s Bay and powered an
ARIANNA station for 40% during the winter [8]. Larger versions of this turbine
are under development to increase uptime and provide autonomous power also at
the RNO-G site in Greenland and the South Pole, which have more challenging
wind conditions.

##  3 The Hexagonal Radio Array - a Test Bed for a Larger Scale Array

The main purpose of ARIANNA was the demonstration of the feasibility of the
in-ice radio technique for detecting ultra-high-energy neutrinos. This was
done in the _hexagonal radio array_ , an array of seven identical radio
detector stations, each comprising four downward-pointing LPDA antennas
arranged in two orthogonal pairs with a separation of $6\text{\,}\mathrm{m}$
[9, 10]. The array was completed in 2015 and operated successfully until it
was decommissioned. ARIANNA demonstrated that an uptime above 90% is
achievable during the summer months when solar power is available [4]. A
neutrino search was performed using the full data set. A straight-forward
template matching analysis yielded no neutrino candidates, with a signal
efficiency of 79%, and a 90% confidence upper limit on the diffuse neutrino
flux we reported of $E^{2}\Phi$ =
$1.7\text{\times}{10}^{-6}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{c}\mathrm{m}^{-2}\mathrm{s}^{-1}\mathrm{s}\mathrm{r}^{-1}$
for a decade-wide logarithmic bin centered at a neutrino energy of
${10}^{18}\text{\,}\mathrm{e}\mathrm{V}$ [4]. The differential limit is shown
in Fig. 2. Although not competitive with existing limits from IceCube or the
Pierre Auger Observatory, the study showed the general feasibility of in-ice
radio neutrino detection.

Figure 1: Sketch of the improved shallow station design. The original four
downward-facing LPDA antennas are complemented by three upward-facing LPDA
antennas for cosmic-ray detection and one dipole to aid background removal and
event reconstruction. Figure 2: Measured astrophysical neutrino flux by
IceCube and limits from different experiments. Also shown is the expected
sensitivity of future detectors and predictions of the GZK neutrino flux based
on measurements of the cosmic-ray flux. ARIANNA-200 was a concept study for a
200 shallow station array on the Ross Ice Shelf.

##  4 Improvements - Towards The Next Generation Neutrino Telescope

Apart from demonstrating the technical feasibility of in-ice neutrino
detection, ARIANNA developed several improvements over the original design
that will increase the sensitivity to UHE neutrinos and improve the
determination of the neutrino’s direction, energy, and flavor.

Upward-facing LPDA antennas were added to the station to measure radio flashes
from cosmic-ray-induced air showers. The DAQ system was extended accordingly
to support eight input channels. This extension serves three purposes: The
rejection of radio signals from air showers that can be a critical background
in a neutrino search, testing the detector capabilities under realistic
conditions (see below in Sec. 6), and enabling cosmic-ray science. To
facilitate the latter, a method was developed to determine the cosmic-ray
direction and energy from data of a single radio detector station [11].

An approx. $10\text{\,}\mathrm{m}$ deep dipole antenna was added to the
station design, which allows measuring two time-delayed signals from the same
in-ice neutrino interaction [4]: One direct trajectory to the antenna and one
trajectory where the signal is reflected off the surface, called D’n’R (short
for Direct and Reflected). This provides a unique signature of in-ice emission
and thus can be used to reject anthropogenic background from above the ice. In
2018, an ARIANNA station with these improvements was installed at the South
Pole, successfully taking data until the station was decommissioned in Jan.
2021. A neutrino search was conducted using the same template-matching
technique as in the previous search of [12]. Using only the four downward-
facing LPDAs, a similar analysis efficiency was obtained as previously on the
Ross Ice Shelf. Using the upward-facing LPDAs and the dipole antenna as
additional discriminators led to a further significant improvement in
background rejection [13]. The final station design is shown in Fig. 1.

The time delay between the direct and reflected signal also allows measuring
the distance to the neutrino interaction, which is a crucial parameter for
estimating the neutrino energy. In a simulation study, we derived the relation
between time delay and distance and studied the corresponding experimental
uncertainties in estimating neutrino energies. We found that the resulting
contribution to the energy resolution is well below the natural limit set by
the unknown inelasticity in the initial neutrino interaction [4]. Furthermore,
if combined with a calibration emitter installed below the ice surface, the
D’n’R time delay can be used to track the snow accumulation throughout the
year. This was demonstrated on the Ross Ice Shelf and provided a precision of
$\mathcal{O}$($1\text{\,}\mathrm{m}\mathrm{m}$) in surface elevation, which is
much better than that needed to apply the D’n’R technique to neutrinos [4]. We
further developed this system into an in-situ calibration system also for the
index-of-refraction profile by adding a second calibration emitter and
determined the optimal positions in a simulation study [14].

Another area of improvement is the trigger to increase the sensitivity to
neutrinos of each single detector station. First, the trigger bandwidth was
optimized [15]: The amplitude of the Askaryan signal increases linearly with
frequency up to a cutoff frequency that depends on the viewing angle but
extends up to $1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ if the shower is
observed close to the Cherenkov angle. However, it was found that the
sensitivity to neutrinos can be increased by up to 50% if the bandwidth is
reduced to the lower end of the frequency band of around
$80-200~{}\mathrm{MHz}$. This is a) because the noise RMS decreases with the
square root of the bandwidth, b) because antennas are typically more sensitive
at low frequencies (the relevant vector effective length is inversely
proportional to the frequency and proportional to the square root of the
gain), and c) because, for a given amplitude at the detector, the geometry of
off-cone events is more favorable where the frequency cutoff quickly drops
into the hundred MHz range.

In addition, sensitivity can be achieved by lowering the trigger threshold.
However, the challenge is that current Askaryan detectors already operate at
such low trigger thresholds so that the vast majority of triggered events are
just unavoidable thermal noise fluctuations. The trigger threshold is then set
by the maximum data rate the detector can handle. To get around these
limitations, we developed a real-time thermal noise rejection algorithm that
enables the trigger thresholds to be lowered, which increases the sensitivity
to neutrinos by up to a factor of two (depending on energy) compared to the
current ARIANNA capabilities [16]. A deep learning discriminator, based on a
Convolutional Neural Network (CNN), was implemented to identify and remove
thermal events in real-time. The network retains 95 percent of the neutrino
signal at a thermal noise rejection factor of ${10}^{5}$. The results of the
simulation study were verified in a lab measurement by feeding in generated
neutrino-like signal pulses and thermal noise directly into the ARIANNA data
acquisition system. More advanced trigger schemes where the antenna signals
are processed continuously are under development, as well as a new all-digital
data acquisition system.

Based on these improvements, a concept study of a 200-station array on the
Ross Ice Shelf was conducted [17], and its expected sensitivity is shown in
Fig. 2. The improved shallow stations are also an integral part of the design
for IceCube-Gen2.

##  5 Simulation and Reconstruction Tools

The ARIANNA initiative has also given rise to the development of innovative
simulation and reconstruction tools. An accurate and adaptable simulation code
is of utmost importance for exploring the capabilities of future detectors, as
well as for the design and evaluation of reconstruction algorithms. The
NuRadioMC simulation code was developed [18], which incorporates our current
best knowledge of the physical processes in a flexible way so that different
detector designs can be simulated quickly. It was shown that the results agree
within a few percent with previous MC codes if the same physics settings were
used. NuRadioMC allows for making reliable predictions on the sensitivity of a
radio detector and can be used for an estimate of reconstruction performance.
The code quickly became the de-facto community standard for in-ice neutrino
detection, with a large community of developers continuously improving and
enhancing its capabilities (see e.g. [19, 20, 21, 22]).

At the same time, the reconstruction framework NuRadioReco [23] was developed
to facilitate the analysis of experimental data and the development of
reconstruction algorithms. Also NuRadioReco was adopted by the community and
serves as the basis for an increasing number of studies. In particular,
NuRadioMC and NuRadioReco enabled the ARIANNA studies described below.

##  6 Development and Test of Reconstruction Techniques

The ability to determine the neutrino’s direction and energy is an important
requirement for in-ice radio detectors. We can’t test the reconstruction
performance directly by comparing it to an alternative detection technique, as
it is done for the radio detection of cosmic rays, because no UHE neutrino has
been measured yet, and large future radio arrays expect to measure just a
handful of neutrinos per year at most. However, what can be done is the
following: Certain aspects of the reconstruction can be verified
experimentally with in-situ tests. This can then be combined with simulation
studies to estimate the reconstruction performance for the expected
distribution of neutrino signals.

The reconstruction of the direction and energy of the neutrino requires the
measurement of the distance to the neutrino vertex, the signal arrival
direction, the viewing angle, and the signal polarization [1]. We
experimentally tested the reconstruction of as many of these parameters as
possible. Using the residual hole from the South Pole Ice Core Project, radio
pulses were emitted from a transmitter located up to
$1.7\text{\,}\mathrm{k}\mathrm{m}$ below the snow surface. By measuring these
signals with an ARIANNA surface station, the signal direction and polarization
reconstruction abilities were quantified [5]. After deconvolving the raw
signals for the detector response and attenuation from propagation through the
ice, the signal pulses show no significant distortion and agree with a
reference measurement of the emitter made in an anechoic chamber. The origin
of the transmitted radio pulse was measured with an angular resolution of
$0.37\text{\,}\mathrm{\SIUnitSymbolDegree}$, indicating that the neutrino
direction can be determined with good precision if the polarization of the
radio pulse can be well determined. In the present study, we obtained a
resolution of the polarization vector of
$2.7\text{\,}\mathrm{\SIUnitSymbolDegree}$. Neither measurement show a
significant offset relative to expectation. One limitation of this study was
that not the complete phase space of possible trajectories through the ice
could be probed, in particular, mostly horizontally propagating signals. More
experimental data probing the effects of propagation through ice on the radio
signals will be useful, in particular, to probe the effects of birefringence
[24, 22].

We developed several complementary reconstruction algorithms and made them
available open-source through NuRadioReco. Among these algorithms, the best
performing one was the forward folding technique [23], where an analytical
model of the emitted radio pulse is modified to account for propagation
effects and compared with the measured signals in all antennas. This allows
for the direct determination of the neutrino direction and shower energy. In a
Monte Carlo study, we applied this technique to a representative data set of
neutrino signals collected in a state-of-the-art shallow detector station
equipped with four downward-facing LPDA antennas and one dipole located at the
South Pole (as shown in Fig. 1). Using all triggered events without any
quality cuts, we achieved an angular resolution of
$3\text{\,}\mathrm{\SIUnitSymbolDegree}$ for the neutrino direction for the
subset of hadronic showers [25, 26]. Additionally, we developed a deep
convolutional neural network to predict the neutrino energy and direction
directly from raw data [27]. The results of this network are encouraging, with
the angular resolution slightly lower than that obtained using the forward
folding technique, but therefore able to reconstruct all event topologies,
including the more complex $\nu_{e}$ charged-current interactions.

Due to the similarities in generated radio signals, cosmic rays are often used
as test beams for neutrino detectors. Some ARIANNA detector stations are
equipped with antennas capable of detecting air showers. We showed – using a
straightforward template-based search – that cosmic rays can be identified,
and that the detection rate agrees with expectations [28], which motivates
that a similar search strategy would work for neutrinos as well.

Since the radio emission properties of air showers are well understood, and
the polarization of the radio signal can be predicted from the arrival
direction, cosmic rays can be used as a proxy to assess the reconstruction
capabilities of the ARIANNA neutrino detector. We used the data of two
stations installed on the Ross Ice Shelf to obtain a sample of cosmic rays and
to reconstruct the polarization of cosmic-ray radio pulses [29]. After
correcting for differences in hardware, the two stations used in this study
showed similar performance in terms of event rate and agreed with simulation.
Subselecting high-quality cosmic rays, the polarizations of these cosmic rays
were reconstructed with a resolution of
$2.5\text{\,}\mathrm{\SIUnitSymbolDegree}$ (68% containment), which agrees
with the expected value obtained from simulation. A large fraction of this
resolution originates from uncertainties in the predicted polarization because
of the contribution of the subdominant Askaryan effect in addition to the
dominant geomagnetic emission. Subselecting events with a zenith angle greater
than $70\text{\,}\mathrm{\SIUnitSymbolDegree}$ removes most influence of the
Askaryan emission, and, with limited statistics, we found the polarization
uncertainty is reduced to $1.3\text{\,}\mathrm{\SIUnitSymbolDegree}$ (68%
containment).

##  7 Outlook

ARIANNA pioneered and paved the way for UHE neutrino detection. With shallow
detector stations on the Ross Ice Shelf and the South Pole, the feasibility of
in-ice radio detection was demonstrated, and the capabilities of the detector
to measure the neutrino direction and energy were tested in in-situ
measurements and using cosmic rays as a test beam. Currently, the Radio
Neutrino Observatory in Greenland is under construction. It will be large
enough to potentially measure the first UHE neutrino. At the same time, an
order of magnitude larger detector is envisioned as part of IceCube-Gen2 at
the South Pole (see Fig. 2) where the improved shallow station design shown in
Fig. 1 is an integral part of the design.

## References

  * [1] S. Barwick and C. Glaser To be published in "Neutrino Physics and Astrophysics", edited by F. W. Stecker, in the Encyclopedia of Cosmology II, edited by G. G. Fazio, World Scientific Publishing Company, Singapore, 2022 (2022) .
  * [2] RNO-G Collaboration, J. A. Aguilar et al. JINST 16 (2021) P03025.
  * [3] IceCube-Gen2 Collaboration, S. Hallmann, B. Clark, C. Glaser, D. Smith, et al. PoS ICRC2021 (2021) 1183.
  * [4] ARIANNA Collaboration, A. Anker et al. JCAP 03 (2020) 053.
  * [5] ARIANNA Collaboration, A. Anker et al. JINST 15 (2020) P09039.
  * [6] TAROGE, ARIANNA Collaboration, S.-H. Wang et al. JCAP 11 (2022) 022.
  * [7] S. A. Kleinfelder, E. Chiem, and T. Prakash, “The SST fully-synchronous multi-ghz analog waveform recorder with nyquist-rate bandwidth and flexible trigger capabilities,” in Proc. of 2014 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC), Seattle, WA, 2014, pp. 1-3.  IEEE, 2014.  1505.07085.
  * [8] Anna Nelles for the ARIANNA collaboration PoS ICRC2019 (2020) 968.
  * [9] ARIANNA Collaboration, S. W. Barwick et al. IEEE Trans. Nucl. Sci. 62 (2015) 2202–2215.
  * [10] ARIANNA Collaboration, S. W. Barwick et al. Astropart. Phys. 70 (2015) 12–26.
  * [11] C. Welling, C. Glaser, and A. Nelles JCAP 10 (2019) 075.
  * [12] ARIANNA Collaboration, A. Anker et al. JCAP 11 (2019) 030.
  * [13] L. Zhao for the ARIANNA collaboration Neutrino 2022 conference .
  * [14] J. Beise and C. Glaser JINST 18 (2023) P01036.
  * [15] C. Glaser and S. W. Barwick PoS ICRC2021 (2021) 1050.
  * [16] ARIANNA Collaboration, A. Anker et al. JINST 17 (2022) P03007.
  * [17] ARIANNA Collaboration, A. Anker et al. arXiv:2004.09841 (2020) .
  * [18] C. Glaser et al. Eur. Phys. J. C 80 (2020) 77.
  * [19] D. García-Fernández, A. Nelles, and C. Glaser Phys. Rev. D 102 (2020) 083011.
  * [20] C. Glaser, D. García-Fernández, and A. Nelles PoS ICRC2021 (2021) 1231.
  * [21] B. Oeyen, I. Plaisier, A. Nelles, C. Glaser, and T. Winchen PoS ICRC2021 (2021) 1027.
  * [22] N. Heyer and C. Glaser Eur. Phys. J. C 83 (2023) 124.
  * [23] C. Glaser, A. Nelles, I. Plaisier, C. Welling, S. W. Barwick, D. García-Fernández, G. Gaswint, R. Lahmann, and C. Persichilli Eur. Phys. J. C 79 (2019) 464.
  * [24] T. M. Jordan, D. Z. Besson, I. Kravchenko, U. Latif, B. Madison, A. Novikov, and A. Shultz Annals Glaciol. 61 (2020) 84.
  * [25] G. Gaswint, “Ph.D dissertation, University of California-Irvine,” 2021.
  * [26] S. Barwick for the ARIANNA collaboration PoS(ICRC2021)1151 .
  * [27] C. Glaser, S. McAleer, S. Stjärnholm, P. Baldi, and S. W. Barwick Astropart. Phys. 145 (2023) 102781.
  * [28] ARIANNA Collaboration, S. W. Barwick et al. Astropart. Phys. 90 (2017) 50–68.
  * [29] ARIANNA Collaboration, A. Anker et al. JCAP 04 (2022) 022.


# A comment on the classical electron self-energy

H. R. de Assis111Corresponding author<EMAIL_ADDRESS>Departamento de Matemática, ICE, Universidade Federal de Juiz de Fora, MG,
Brazil  B. F. Rizzuti222Contact<EMAIL_ADDRESS>Departamento de
Física, ICE, Universidade Federal de Juiz de Fora, MG, Brazil

###### Abstract

This paper is devoted to the analysis of the divergence of the electron self-
energy in classical electrodynamics. To do so, we appeal to the theory of
distributions and a method for obtaining corresponding extensions. At first
sight, electrostatics implies a divergence once we treat the electron as a
charged point particle. However, our construction shows that its self-energy
turns out to be an undetermined constant upon renormalization. Appealing to
empirical results we may fix its value, demanding, for example, that all its
mass comes from an electrostatic origin.

Keywords: Theory of distributions. Extension of distributions. Electron self-
energy.

##  1 Introduction

One of the most compatible matches between theory and experiment in physics
can be seen in the field of Quantum Electrodynamics (QED), as the precision on
the electron magnetic moment goes far from expected [1]. Since the early days
of QED computations, it became clear that one would have to deal with
singularities, related, for example, with self-interactions. In turn, this has
led the community to stare at fields not as maps, but as distributions. In the
case of the electric field $\vec{E}(\vec{x},t)$ originated from a point
particle, for instance, one would expect an ultraviolet divergence at origin,
while $\int d^{3}\vec{x}dt\vec{E}(\vec{x},t)f(\vec{x})=\vec{E}(f)$ is well
behaved [2]. Here, $f(\cdot)$ is a smooth function of compact support. We are
interested, in this manuscript, on the self-energy of the electron. While it
has a fascinating history so depicted in [3], involving an entire war and a
new generation of physicists developing regularization and renormalization
techniques, the classical counterpart is often subdue, justifying our approach
here.

Simply put, the self-energy of a charged particle, such as the electron, is
the measure of the energy it posses when freed from any other interaction, be
it with other particles or with given fields. One finds in the study of
classical electrodynamics that, summed to the kinetic and potential energies a
given collection of particles might have, a system composed of charged
particles has a quantity of energy related to the electromagnetic field it
generates [4].

The electromagnetic system we wish to examine could be seen, at first, as the
simplest case: that of an electron, stationary, freed from any other
interaction. Seen as a point particle \- that is, supposing it has no internal
structure and being solely described by the position in which its whole charge
is stored - which is the standard way one encounters at first [4, 5], the
electric field and potential are given by

$\textbf{E}(\textbf{r})=\frac{1}{4\pi\epsilon_{0}}\frac{e}{r^{2}}\hat{\textbf{r}},\qquad
V(\textbf{r})=\frac{1}{4\pi\epsilon_{0}}\frac{e}{r},$ (1)

where $e$ denotes the strength of its charge.

Meanwhile, the expression for the self-energy for a system with electric field
E and magnetic field B is

$E=\frac{\epsilon_{0}}{2}\int_{\mathds{R}^{3}}\left(\textbf{E}^{2}+c^{2}\textbf{B}^{2}\right)d\tau,$
(2)

so that, using (1), we obtain

$E_{0}=\frac{\epsilon_{0}}{2}\int_{\mathds{R}^{3}}\left(\frac{1}{4\pi\epsilon_{0}}\right)^{2}\left(\frac{e}{r^{2}}\right)^{2}d\tau.$

We denote $E_{0}$ the self-energy of interest here, as we are allegedly
neglecting the magnetic field due to our interest only in the static case.
Using spherical coordinates,333Since different materials might use different
notations concerning the polar coordinates $\theta$ and $\phi$, we make
explicit that we are considering here $\begin{cases}x=r\hskip
2.0pt\mathrm{sen}{\theta}\cos{\phi},\\\ y=r\hskip
2.0pt\mathrm{sen}{\theta}\hskip 2.0pt\mathrm{sen}{\phi},\\\
z=r\cos{\theta},\end{cases}$ where $\theta\in[0,\pi]$, $\phi\in[0,2\pi)$.

$\begin{split}E_{0}&=\frac{e^{2}}{(4\pi\epsilon_{0})^{2}}\left(\int_{0}^{2\pi}d\phi\right)\left(\int_{0}^{\pi}\hskip
2.0pt\mathrm{sen}{(\theta)}d\theta\right)\left(\int_{0}^{+\infty}\frac{1}{r^{2}}dr\right)\\\
&=\frac{e^{2}}{8\pi\epsilon_{0}}\left[\frac{1}{r}\right]_{+\infty}^{0}\\\
&=+\infty.\end{split}$ (3)

The conclusion we arrive then is that there is an infinite amount of energy
stored in the field of a simple electron positioned at the origin of our
system, if one considers it to be a stationary point particle. Needless to
say, any satisfactory field theory, both classical or quantum, must resolve
such type of divergences. Should we discard the assumption that the electron
has no spatial extension? Both theoretical and experimental results seem to
point in an opposite way [6, 7], indicating that we should seek a improvement
in the theory and in the conception of self-energy itself.

Therefore, here we present one of the available methods for the “removal” of
such infinite quantities, a process known as renormalization [8]. The main
idea of renormalization is that these infinities can be justified by
attributing them to quantities which we cannot directly measure (something
that can be seen as a parallel with the acceptance of complex numbers in the
formalism of quantum mechanics). Take, for example, our case of the electron
and its infinite self-energy. In calculating $E_{0}$, we are, simultaneously,
calculating the mass $m_{elec}$ arising from the electric field of such
particle, since Einstein’s relativity theory affirms that mass and energy are
but two manifestations of the same phenomenon. In light of this, we conclude
that the divergence of $E_{0}$ implies that the electron possesses an infinite
inertia. If, however, we assume there exists another contribution for the
effective mass (that is, the one we can actually measure), originated from
some unknown effect other than electromagnetism, then we might conceive that
this new contribution is negative enough to oppose the infinity appearing from
$m_{elec}$. The hypothesis of another source contributing for the effective
mass is not something hardly justified, since we know that neutral bodies also
have mass and generate no electric or magnetic field. Thus, assuming this new
contribution for the mass of the particle, independent of where it comes from,
we can “erase” the infinite we have just found, obtaining the so called mass
renormalization of the electron. It might be difficult to accept a so far
unknown (non-electromagnetic) interaction which is responsible for a strong
renormalization of the mass, without leaving a fingerprint in the dynamical
particle sector. Our effective result shows that a finite mass/energy is
obtained, though, even without invoking any other renormalization procedure.

Such a primitive method for renormalization gave rise to new and more advanced
approaches which came to be used in the renormalization of other infinite
quantities, specially in Quantum Field Theory (QFT), which advanced quite a
lot in the decades following the emergence of quantum mechanics and which
presented similar problems with divergent integrals in its equations [9]. For
such a new theory, the methods had to be refined in its mathematical
formulation and, meanwhile, the development of approaches such as constructive
quantum field theory [2] and causal perturbation theory (CPT) [10, 11] made
clear that distribution theory was a crucial subject for the understanding of
such new ideas. It was also through this theory that it was perceived the
connection between the appearance of divergent quantities and the product of
distributions, whereas distribution theory is strictly linear, as we shall
see. This obstacle can be overcome using the idea of extension of
distributions, which we will encounter in Sec. 2. What we shall present here
is a direct consequence of the work of Epstein and Glaser [11] and texts which
adapted their ideas [12, 13, 14, 15]. This renormalization procedure is
equivalent to the extension of distributions and this is exactly our approach
in this paper. Curiously enough, just after the CPT development there were
only but a couple of applications, probably due to the necessity of the
rigorous functional analysis so applied [16]. Currently, it is more than
established. Throughout the method, the renormalization is established by
subtracting pertinent test functions, while the physicist community is used to
working with distributions in an integral kernel representation. Our work
shows (i) that not only does distributions present themselves as good
candidates for describing fields, but it also (ii) discusses how a particular
scheme of renormalization can be accomplished.

There are different methods for literally “sweeping the infinities under the
rug” [8]. In the first step, one regularizes the divergent integral, rendering
it finite, suppressing the UV physics by a regulator. Then, one posits
renormalization conditions, imposing finite conditions on physical quantities.
In turn, this defines the cutoff dependence of the bare parameters. After
evaluating the now solvable integral, the cutoff parameter is sent to
infinity. The result depends of a finite term independent of the regulator
plus another one that blows up as the parameter increases. This result is
confusing but not that tragic, after all we can only measure the physical
values, allegedly finite. The process of the CPT renormalization is managed to
take place only once in every step. The lower-order contributions are already
renormalized. This corresponds to finding all divergent subgraphs in the
traditional approach and simplifies the proof of the same construction to all
orders [16].

In view of this, this paper expresses how distributions are key elements even
within the classical regime of electrodynamics. We shall also present, in Sec.
2 a method for extending distributions. Finally, we apply such method, in Sec.
3, to the renormalization of the self-energy of the electron, which is the
main objective of our paper. We point out that, although distributions do not
have direct physical meaning, the values that we find upon their application
on smooth functions are clearly physical and finite. In particular, the value
to the electron self-energy would not have been achieved without the use of
distributions (and their extensions). All this chain was written to be an
instructive introduction to the theme. The basic notions about distribution
theory, though, are assumed to be known and the interested reader may view a
deeper discussion in the existing literature [20, 17, 18, 19, 21].

##  2 Preliminary definitions and results

As we have hinted in Sec. 1, the developments of QFT showed that
renormalization in causal perturbation theory depends heavily on distribution
theory, more specifically on the procedures for obtaining extensions of
distributions whose behavior at the origin (such nomenclature will become
clear later) does not allow that we apply these over functions whose support
contains the origin. Our main problem becomes:

Given a distribution $T_{0}$ which is only well defined when we apply it on
test functions $\varphi\in\mathcal{D}(\mathds{R}^{n})$ for which
$0\notin\textit{supp }\varphi$, how can we construct a new distribution
$T\in\mathcal{D}^{\prime}(\mathds{R}^{n})$ which is the extension of $T_{0}$,
that is, such that $\left\langle T_{0},\varphi\right\rangle=\left\langle
T,\varphi\right\rangle$ whenever $0\notin\textit{supp }\varphi$?

A presentation of the method for constructing such extensions is the goal of
this section. The ideas here presented originate mainly from [16, 13, 12] and
references therein.

###  2.1 Extensions of distributions

Consider a regular distribution $T_{f}=f\in\mathcal{D}(\mathds{R}^{n})$ such
that $f(x)=0$ for every $x$ belonging to a subset $U\subset\mathds{R}^{n}$. It
is then evident that, for all $\varphi\in\mathcal{D}(\mathds{R}^{n})$ such
that $\textit{supp }\varphi\subset U$ (we write, in this case,
$\varphi\in\mathcal{D}(U)$), we have $\left\langle
f,\varphi\right\rangle=f(\varphi)=0$. Note that this can be a form of
characterizing the support $\textit{supp }f$ of the function $f$. With the
intent of extending this notion to distributions
$T\in\mathcal{D}^{\prime}(\mathds{R}^{n})$, we say that $T$ is zero in a
subset $U\subset\mathds{R}^{n}$ when

$\left\langle
T,\varphi\right\rangle=0\;,\quad\forall\;\varphi\in\mathcal{D}(U).$

We are thus able to define the support of a distribution $T$.

######  Definition 2.1.

For a given $T\in\mathcal{D}^{\prime}(\mathds{R}^{n})$, the support of $T$ is
the subset $\textit{supp }{T}\subset\mathds{R}^{n}$ given by

$\textit{supp }{T}:=\\{x\in\mathds{R}^{n}\;;\;x\mbox{ does not contain a
neighborhood in which $T$ is zero}\\}.$

In an equivalent manner, $\textit{supp }{T}$ can be seen as the complement of
the biggest subset in which $T$ is zero.

This definition, in turn, allows one to define an important subspace of
$\mathcal{D}^{\prime}(\mathds{R}^{n})$. For any subset
$U\subset\mathds{R}^{n}$, the subspace of
$\mathcal{D}^{\prime}(\mathds{R}^{n})$ given by the distributions such that
$\textit{supp }{T}\subset U$ will be denoted by $\mathcal{D}^{\prime}(U)$.

One important example can be given by the Dirac delta distribution, for which
we have $\textit{supp }{\delta_{x_{0}}}=\\{x_{0}\\}$. We can actually obtain a
reciprocate from this result with the following lemma (for the proof, see for
example [18]).

######  Lemma 2.1.

If $T\in\mathcal{D}^{\prime}(\mathds{R}^{n})$ is such that $\textit{supp
}T=\\{x_{0}\\}$, then there exists $m\in\mathds{N}$ and constants $c_{\nu}$,
$\left|\nu\right|\leq m$, such that

$T=\sum_{\left|\nu\right|\leq m}c_{\nu}D^{\nu}\delta_{x_{0}}.$ (4)

In what follows, we define a quantity which probes the behavior of a
distribution $T$ in the origin, in terms of singularities. Since we have drawn
a clear distinction between bona fide functions (regular distributions) and
general (singular) distributions, we must develop for the latter the notion of
studying the behavior of $T$ at some point in $\mathds{R}^{n}$. It is in this
ground that we introduce the concept of a pull-back.

######  Definition 2.2.

The pull-back of $T\in\mathcal{D}^{\prime}(\mathds{R}^{n})$ over an
invertible444The use of $\Phi$ invertible is a particular case of a definition
that can be made more general. More details can be found in [18].
transformation $\Phi:\mathds{R}^{n}\rightarrow\mathds{R}^{n}$ is a new
distribution $\Phi^{*}T$, defined by555Here the symbol $\left|DF(y)\right|$
represents the Jacobian of $F:\mathds{R}^{n}\longrightarrow\mathds{R}^{n}$.

$\left\langle\Phi^{*}T,\varphi\right\rangle=\left\langle
T,\left|D\Theta(y)\right|\Theta^{*}\varphi\right\rangle\;,\forall\;\varphi\in\mathcal{D}(\mathds{R}^{n}),$
(5)

where $\Theta=\Phi^{-1}$. It is common the notation $T(\Phi(x))$ (which we
shall employ from here on, for conformity) for the distribution $\Phi^{*}T$,
making reference to the definition of the pull-back of a function.

######  Remark 2.1.

Let us take here some more lines to achieve a better understanding of
Definition 2.2. Despite the indication given by the notation $T(\Phi(x))$, the
pull-back of a distribution should not be read as an ordinary function. If $T$
is singular, then $\Phi^{*}T$ is only well defined as a new distribution,
which will again be singular. Thus, we should not interpret $T(\Phi(x))$ as an
object which varies with $x\in\mathds{R}^{n}$, but instead as a distribution
dependent on the transformation $\Phi$. Nonetheless, when regarding simple
(but important) cases, such as $\Phi(x)=\lambda x$, it is easier and perhaps
more didactic to express the transformation directly as the argument of $T$.

After such remarks, we present the following definition.

######  Definition 2.3.

Let $T\in\mathcal{D}^{\prime}(\mathds{R}^{n})$ be a distribution. The scaling
degree of $T$ is the real number (or $\pm\infty$), denoted here by
$\sigma(T)$, such that

$\sigma(T)=\inf\\{s\in\mathds{R}\;;\;\lambda^{s}T(\lambda
x)\xrightarrow{\lambda\to 0^{+}}0\\}.$

The singular order of $T$, denoted by $\omega(T)$, is the value

$\omega(T)=[\sigma(T)]-n,$

where $[m]$ denotes the biggest integer smaller (or equal) to $m$.

######  Example 2.1.

It is instructive to pass through some examples of distributions and their
respective scaling degrees, in the hopes of making our definitions clearer.
The examples will be useful to our further discussions as well.

  1. 1.

If $T=\delta\in\mathcal{D}^{\prime}(\mathds{R})$, then $\sigma(\delta)=1$.
This is given by the result $\delta(\lambda x)=\lambda^{-1}\delta(x)$, which
follows directly from (5). More generally, dealing with the $n$-dimensional
case, $\delta\in\mathcal{D}^{\prime}(\mathds{R}^{n})$, we have

$\left|D\Phi^{-1}(y)\right|=\lambda^{-n},\mbox{ if }\Phi(x)=\lambda x,$

implying that

$\left\langle\delta(\lambda
x),\varphi\right\rangle=\lambda^{-n}\left\langle\delta,\varphi\right\rangle.$

This means that $\sigma(\delta)=n$.

  2. 2.

If $f$ is a continuous function, of one variable, homogeneous with degree $m$,
meaning $f(\lambda x)=\lambda^{m}f(x)$, then

$\left\langle\lambda^{s}f(\lambda
x),\varphi\right\rangle=\lambda^{s}\int_{-\infty}^{+\infty}\lambda^{m}f(x)\varphi(x)dx=\lambda^{s+m}\left\langle
f,\varphi\right\rangle.$

It follows immediately from this that $\sigma(f)=-m$. Thus, the scaling degree
of a homogeneous function is the (additive) inverse of its homogeneity degree.

We now posses the appropriate tools for enunciating the results concerning
proper extensions of distributions. We must separate our problem into two
cases, which are characterized by $\omega(T)<0$ and $\omega(T)\geq 0$, that
is, the order relation of the scaling degree and the dimension of our space.
The biggest difference between the two situations is in the uniqueness of the
extension.

######  Theorem 2.1.

Let $T_{0}\in\mathcal{D}^{\prime}(\mathds{R}^{n}\backslash\\{0\\})$ such that
$\sigma(T_{0})=s<n$. Then there exists a unique distribution
$T\in\mathcal{D}^{\prime}(\mathds{R}^{n})$ such that $\sigma(T)=s$ and

$\left\langle T,\varphi\right\rangle=\left\langle
T_{0},\varphi\right\rangle\;,\quad\forall\;\varphi\in\mathcal{D}(\mathds{R}^{n}\backslash\\{0\\}).$

Before passing to the next (and arguably more important) result, we must
define some more concepts important to our study. We will denote by
$\mathcal{D}_{\omega}(\mathds{R}^{n})$ the subspace of
$\mathcal{D}(\mathds{R}^{n})$ composed of functions such that their
derivatives up to order $\omega$ vanish in the origin. Thus, for some natural
number $\omega>0$,

$\mathcal{D}_{\omega}(\mathds{R}^{n})=\\{\varphi\in\mathcal{D}(\mathds{R}^{n})\;;D^{\alpha}\varphi(0)=0\;,\left|\alpha\right|\leq\omega\\}.$

Now, for every function $\varphi\in\mathcal{D}(\mathds{R}^{n})$, its Taylor
expansion will be given by

$\varphi(x)=\sum_{\nu}\frac{x^{\nu}}{\nu!}D^{\nu}\varphi(0),$ (6)

from which, separating the terms whose multi-index have norm
$\left|\nu\right|>\omega$, we obtain

$\varphi(x)=\sum_{\left|\nu\right|\leq\omega}\frac{x^{\nu}}{\nu!}D^{\nu}\varphi(0)+\sum_{\left|\nu\right|>\omega}\frac{x^{\nu}}{\nu!}D^{\nu}\varphi(0).$
(7)

Therefore, the inclusion $\varphi\in\mathcal{D}_{\omega}(\mathds{R}^{n})$ is
equivalent to saying that the first summation in (7) is zero.

The projection of $\mathcal{D}(\mathds{R}^{n})$ onto
$\mathcal{D}_{\omega}(\mathds{R}^{n})$ also plays a crucial role to our second
result. Let $\mathcal{W}=\mathcal{D}_{\omega}(\mathds{R}^{n})^{\bot}$ be the
orthogonal complement of $\mathcal{D}_{\omega}(\mathds{R}^{n})$ so that
$\mathcal{D}(\mathds{R}^{n})=\mathcal{D}_{\omega}(\mathds{R}^{n})\oplus\mathcal{W}$.777$\mathcal{D}_{\omega}(\mathds{R}^{n})$
is evidently closed due to the continuity of any $D^{\alpha}\varphi$. This
allows us to write functionals in $\mathcal{D}^{\prime}(\mathds{R}^{n})$ as

$T=T_{\omega}\oplus l\;,\quad
T_{\omega}\in\mathcal{D}_{\omega}^{\prime}(\mathds{R}^{n})\;,\quad
l\in\mathcal{W}^{\prime}.$

We should ask ourselves then what are the functionals in
$\mathcal{W}^{\prime}$. They are characterized as elements
$l\in\mathcal{D}^{\prime}(\mathds{R}^{n})$ such that $\left\langle
l,\varphi\right\rangle=0$ for any
$\varphi\in\mathcal{D}_{\omega}(\mathds{R}^{n})$. More specifically, we have
the following.

######  Lemma 2.2.

For an arbitrary real number $\omega>0$, we have

$\mathcal{W}^{\prime}=\left\\{l\in\mathcal{D}^{\prime}(\mathds{R}^{n})\;;\;l=\sum_{\left|\alpha\right|\leq\omega}c_{\alpha}D^{\alpha}\delta\;,\quad
c_{\alpha}\in\mathds{C}\right\\}.$

Once $\mathcal{W}^{\prime}$ is a linear space, it is a consequence of Lemma
2.2 that $\mathcal{B}=\\{D^{\alpha}\delta\;;\left|\alpha\right|\leq\omega\\}$
is a basis to such dual space. We use, however, a different set of
distributions to represent the orthogonal projection of
$\mathcal{D}^{\prime}(\mathds{R}^{n})$ on
$\mathcal{D}_{\omega}^{\prime}(\mathds{R}^{n})$, one that depends upon a
particular test function $w\in\mathcal{D}(\mathds{R}^{n})$. We can show that,
for any function $w\in\mathcal{D}(\mathds{R}^{n})$ such that $w(0)\neq 0$, the
set888This means the action of an element of $\mathcal{C}$ over a
$\varphi\in\mathcal{D}(\mathds{R}^{n})$ is $\left\langle
D^{\alpha}\delta(w^{-1}\cdot),\varphi\right\rangle=\left\langle
D^{\alpha}\delta,w^{-1}\varphi\right\rangle$.
$\mathcal{C}=\\{D^{\alpha}\delta(w^{-1}\cdot)\;;\left|\alpha\right|\leq\omega\\}$
is a basis to $\mathcal{W}^{\prime}$ as well.

The proof of this argument proceeds as follows: the number of elements in
$\mathcal{C}$ equals the number of elements in $\mathcal{B}$, so that one need
only to show that the elements of the latter may be generated by
$\mathcal{C}$. For this we use the Leibniz rule, which implies that, for any
multi-index $\alpha$ and any $\psi\in\mathcal{D}(\mathds{R}^{n})$ that does
not vanish in a given point $x\in\mathds{R}^{n}$,

$D^{\alpha}\varphi(x)=\frac{1}{\psi(x)}D^{\alpha}(\psi\varphi)(x)-\frac{1}{\psi(x)}\sum_{0\leq\beta<\alpha}\frac{\alpha!}{\beta!(\alpha-\beta)!}(D^{\beta}\varphi)(x)(D^{\alpha-\beta}\psi)(x).$

Hence, if $\left\langle
D^{\beta}\delta,\varphi\right\rangle=(-1)^{\left|\beta\right|}D^{\beta}\varphi(0)$
may be written as a linear combination of terms such as $\left\langle
D^{\gamma}\delta(w^{-1}\cdot),\varphi\right\rangle=(-1)^{\left|\gamma\right|}D^{\gamma}(w^{-1}\varphi)(0)$
for arbitrary999This inequality means that $\beta_{i}\leq\alpha_{i}$ for all
$i\in\\{1,\cdots,n\\}$ and at least one of its coordinates, $\beta_{j}$, is
strictly lesser than $\alpha_{i}$. $\beta<\alpha$, then the same holds for
$D^{\alpha}\varphi(0)$ by an argument of induction. We conclude our proof
noting that the result is valid for $\alpha=(0,\cdots,0)$,

$\left\langle
D^{\alpha}\delta,\varphi\right\rangle=\varphi(0)=w(0)\frac{\varphi(0)}{w(0)}=w(0)\left\langle
D^{\alpha}\delta(w^{-1}\cdot),\varphi\right\rangle.$

Just as any element in $\mathcal{B}$ may be written as a linear combination of
elements in $\mathcal{C}$, the same is true for $\mathcal{W}^{\prime}$. In
addition, the set
$\mathcal{E}=\\{\frac{(-1)^{\left|\alpha\right|}}{\alpha!}w(x)x^{\alpha}\;;\left|\alpha\right|\leq\omega\\}$
generates $\mathcal{W}$ and will be a basis to the dual of $\mathcal{C}$,
since101010The Kroenecker delta used here is a generalization for multiple
variables, which is $1$ whenever $\alpha=\beta$, and zero otherwise.

$\left\langle
D^{\alpha}\delta(w^{-1}\cdot),\frac{(-1)^{\left|\beta\right|}}{\beta!}w(x)x^{\beta}\right\rangle=\frac{(-1)^{\left|\alpha\right|+\left|\beta\right|}}{\beta!}(D^{\alpha}x^{\beta})(0)=\delta_{\alpha,\beta}.$

Then, we can write the projection operator of $\mathcal{D}(\mathds{R}^{n})$
onto $\mathcal{D}_{\omega}(\mathds{R}^{n})$ in the form

$\displaystyle W_{(\omega;w)}\colon\mathcal{D}(\mathds{R}^{n})$
$\displaystyle\longrightarrow\mathcal{D}_{\omega}(\mathds{R}^{n})$
$\displaystyle\varphi(x)$
$\displaystyle\longmapsto\varphi(x)-w(x)\sum_{\left|\alpha\right|\leq\omega}\frac{x^{\alpha}}{\alpha!}\left(D^{\alpha}\frac{\varphi}{w}\right)(0),$
(8)

for any $w\in\mathcal{D}(\mathds{R}^{n})$ such that $w(0)\neq 0$.

We chose $w^{-1}$ instead of the function $w$ itself because the former allows
us to write an interesting and useful property, to be used ahead. Namely, the
operator $W$ satisfies

$W_{(\omega;w)}(w\varphi)=wW_{(\omega;1)}(\varphi).$ (9)

We point out that there is no problem at all when using $W_{(\omega;\psi)}$,
with $\psi$ being the constant function equal to unity. Although
$W_{(\omega;1)}\varphi$ is not a test function (its support is not compact),
it is infinitely differentiable and, hence, its product with $w$ will be, in
fact, an element of $\mathcal{D}(\mathds{R}^{n})$.

In accordance with (9), we have, if $\left|\alpha\right|\leq\omega$,

$W_{(\omega;w)}(wx^{\alpha})=wW_{(\omega;1)}x^{\alpha},$

that is,

$W_{(\omega;w)}(wx^{\alpha})(x)=w(x)\left[x^{\alpha}-\sum_{\left|\beta\right|\leq\omega}\frac{x^{\beta}}{\beta!}\left(D^{\beta}x^{\alpha}\right)(0)\right].$

The last sum reduces to $x^{\alpha}$, so that

$W_{(\omega;w)}(wx^{\alpha})\equiv 0.$ (10)

Our next result is the following

######  Theorem 2.2.

Let $T_{0}\in\mathcal{D}^{\prime}(\mathds{R}^{n}\backslash\\{0\\})$ such that
$\sigma(T_{0})=s\geq n$ and $\omega=\omega(T_{0})=s-n$. Moreover, given
$w\in\mathcal{D}(\mathds{R}^{n})$, with $w(0)\neq 0$, and constants
$C^{\alpha}\in\mathds{C}$ for all multi-index $\alpha$, with
$\left|\alpha\right|\leq\omega$. There exists one, and only one, distribution
$T\in\mathcal{D}^{\prime}(\mathds{R}^{n})$ such that $\sigma(T)=s$ and
satisfying

  1. 1.

$\left\langle T,\varphi\right\rangle=\left\langle
T_{0},\varphi\right\rangle\;,\quad\forall\;\varphi\in\mathcal{D}(\mathds{R}^{n}\backslash\\{0\\})$.

  2. 2.

$\left\langle T,wx^{\alpha}\right\rangle=C^{\alpha}$.

Specifically, $T$ is given by

$\left\langle T,\varphi\right\rangle=\left\langle
T_{\omega},W_{(\omega;w)}\varphi\right\rangle+\sum_{\left|\alpha\right|\leq\omega}\frac{C^{\alpha}}{\alpha!}\left(D^{\alpha}\frac{\varphi}{w}\right)(0),$
(11)

where $T_{\omega}$ is the only extension guaranteed by the Theorem 2.1 and
$W_{(\omega;w)}$ is the operator $W$, defined in (8).

######  Remark 2.2.

We can simplify even further our calculations of the extension of $T_{0}$ if
we restrict the class of functions permitted to $w$. More specifically, if we
take $w$ such that $(D^{\alpha}w)(0)=\delta_{\alpha,0}$, which is equivalent
to taking $w$ equal to 1 in a neighborhood of the origin, we have

$\begin{split}(D^{\alpha}\frac{\varphi}{w})(0)&=\sum_{\begin{subarray}{c}0\leq\beta\leq\alpha\\\
\beta\neq\alpha\end{subarray}}\frac{\alpha!}{\beta!(\alpha-\beta)!}(D^{\beta}\varphi)(0)(D^{\alpha-\beta}w^{-1})(0)\\\
&=D^{\alpha}\varphi(0),\end{split}$ (12)

since any derivation of $w^{-1}$ will also carry some derivative of $w$
itself. From that, it follows that $W$ reduces to

$(W_{(\omega;w)}\varphi)(x)=\varphi(x)-w(x)\sum_{\left|\alpha\right|\leq\omega}\frac{x^{\alpha}}{\alpha!}D^{\alpha}\varphi(0)$

and the extension $T$ given by Theorem 2.2 can, therefore, be written as

$\left\langle T,\varphi\right\rangle=\left\langle
T_{\omega},W_{(\omega;w)}\varphi\right\rangle+\sum_{\left|\alpha\right|\leq\omega}\frac{C^{\alpha}}{\alpha!}\left(D^{\alpha}\varphi\right)(0).$
(13)

Such functions, satisfying $(D^{\alpha}w)(0)=\delta_{\alpha,0}$, are called
Epstein-Glaser functions (see, for example, [14]).

###  2.2 Dependence of the extension on the test function $w(x)$

We have seen that, for the case when $\sigma(T_{0})\geq n$, it does not seem
possible to get rid of the dependence of the extension
$T\in\mathcal{D}^{\prime}(\mathds{R}^{n})$ on the test function
$w\in\mathcal{D}(\mathds{R}^{n})$ chosen to construct the projection of
$\mathcal{D}(\mathds{R}^{n})$ over $\mathcal{D}_{\omega}(\mathds{R}^{n})$. We
can, nonetheless, observe the behavior of this dependence, mainly by studying
the term $\left\langle T_{\omega},W_{(\omega;w)}\varphi\right\rangle$, which
we denote, following [16], the integral kernel of the extension. Here, we will
stick with the supposition that $w$ is an Epstein-Glaser function, in the
sense defined above.

Thus, choosing two Epstein-Glaser test functions
$w_{1},w_{2}\in\mathcal{D}(\mathds{R}^{n})$, we have

$\begin{split}(W_{(\omega;w_{1})}\varphi)(x)&=\varphi(x)-w_{1}(x)\sum_{\left|\alpha\right|\leq\omega}\frac{x^{\alpha}}{\alpha!}D^{\alpha}\varphi(0)\\\
&=\varphi(x)-w_{2}(x)\sum_{\left|\alpha\right|\leq\omega}\frac{x^{\alpha}}{\alpha!}D^{\alpha}\varphi(0)+(w_{2}(x)-w_{1}(x))\sum_{\left|\alpha\right|\leq\omega}\frac{x^{\alpha}}{\alpha!}D^{\alpha}\varphi(0)\\\
&=(W_{(\omega;w_{2})}\varphi)(x)+(w_{2}(x)-w_{1}(x))\sum_{\left|\alpha\right|\leq\omega}\frac{x^{\alpha}}{\alpha!}D^{\alpha}\varphi(0)\end{split}$
(14)

and, applying $T_{\omega}$ over this expression, we obtain (bearing in mind
that $T_{\omega}$ is unique, by Theorem 2.1)

$\left\langle T_{\omega},W_{(\omega;w_{1})}\varphi\right\rangle=\left\langle
T_{\omega},W_{(\omega;w_{2})}\varphi\right\rangle+\sum_{\left|\alpha\right|\leq\omega}\left\langle
T_{\omega},(w_{2}(x)-w_{1}(x))\frac{x^{\alpha}}{\alpha!}\right\rangle
D^{\alpha}\varphi(0).$ (15)

We therefore conclude that the application of $T_{\omega}$ over different
projections differ only by a linear combination of terms of the form
$\left\langle D^{\alpha}\delta,\varphi\right\rangle$, that is, by application,
over the test function $\varphi$, of operators belonging to
$\mathcal{W}^{\prime}$.

Our goal ahead will be, however, to get rid of the restriction that $w$ be a
test function. As we have already mentioned, this seems to be a crucial
condition for defining $W$ as a projection operator, since it is responsible
for the fact that the term
$w(x)\sum_{\left|\alpha\right|\leq\omega}\frac{x^{\alpha}}{\alpha!}D^{\alpha}\varphi(0)$
has compact support. However, very often (as will be the case in a while) a
distribution which is not well behaved at the origin will behave nicely at
infinity. To be more precise, we mean that such distributions will be well
defined when applied to functions $\varphi\in C^{\infty}(\mathds{R}^{n})$
whose support may not be compact. In that sense, taking a sequence
$(w_{k})\subset\mathcal{D}(\mathds{R}^{n})$ whose pointwise limit $w$ is a
function111111Some works, such as [16], go even further as to only ask that
$w\in\mathcal{D}^{\prime}(\mathds{R}^{n})$ be a distribution. We will not need
this generality, so that we have preferred to omit this possibility. in
$C^{\infty}(\mathds{R}^{n})$ and $T_{0}$ is such that

$\lim_{k\to\infty}\left\langle
T_{\omega},W_{(\omega;w_{k})}\varphi\right\rangle\in\mathds{C},\quad\forall\quad\varphi\in\mathcal{D}(\mathds{R}^{n}),$

then there is no motive to not consider $w$ in our renormalization scheme. We
shall see how this is an important part for the application of this method of
extending distributions in the following section.

##  3 Application to the electron self-energy

We can now finally attack the electron self-energy problem, seen as a point
particle. We now posses sufficient machinery to see this task as a pathology
to be faced with extension of distributions defined over
$\mathcal{D}(\mathds{R}^{n}\backslash\\{0\\})$. To clarify our aims, we will
translate the issues of electrostatics to the language of distributions.

The first and default example is the charge distribution of an electron, seen
as a charged point particle, which is represented by the Dirac delta
$\delta(\cdot)$, centered where we suppose the whole electric charge of the
particle should be concentrated. Actually, this particular case is not the
only one where we consider the charge distribution $\rho$ as a generalized
function. After all, it would be an incredible coincidence in nomenclature if
the charge distributions presented in the electrodynamics were not presented
by distributions $\rho\in\mathcal{D}^{\prime}(\mathds{R}^{3})$. This is one of
the main contributions of the theory of generalized functions to
electrodynamics. Not only does it incorporate more general charge
distributions $\rho$ (which could not be defined just as real functions in
$\mathds{R}^{3}$), but it also eases some of its manipulations.

######  Example 3.1.

To illustrate our last paragraph, let us analyze the representation of an
electric dipole as a distribution in $\mathcal{D}(\mathds{R}^{3})$. The pure
dipole, considered an idealization such as point charges, is constructed as
follows: start by setting two charges, $q$ and $-q$, separated by a fixed
distance $\varepsilon$. Suppose that they lie in the $x$ coordinate, with $-q$
at the origin and $q$ at $x=\varepsilon$, accordingly. That way, the
distribution $\rho\in\mathcal{D}(\mathds{R})$ will be

$\rho=-q\delta_{0}+q\delta_{\varepsilon}=q(\delta_{\varepsilon}-\delta_{0}).$

If we set $q=1/\varepsilon$, then, in the limit $\varepsilon\to 0^{+}$ we
obtain a distribution whose total charge vanishes. This result is not correct
nonetheless. A null charge distribution would lead to a zero electric field,
which is not the case for a dipole (see [4] for the expression of $\vec{E}$ in
this case).

This simple argument indicates that the distribution formalism is more than
necessary to an accurate description of even the basics of electrostatics.

Now, we apply
$\rho_{\varepsilon}=\varepsilon^{-1}(\delta_{\varepsilon}-\delta_{0})$ over
any $\varphi\in\mathcal{D}(\mathds{R})$. We have

$\left\langle\rho_{\varepsilon},\varphi\right\rangle=\frac{1}{\varepsilon}\left\langle\delta_{\varepsilon}-\delta_{0},\varphi\right\rangle=\frac{\varphi(\varepsilon)-\varphi(0)}{\varepsilon}$

and, therefore, we shall obtain, in the limit $\varepsilon\to 0^{+}$,

$\left\langle\rho_{\varepsilon},\varphi\right\rangle\to\varphi^{\prime}(0)=-\left\langle\delta^{\prime},\varphi\right\rangle,$

that is,

$\rho_{dip}=-\delta^{\prime}.$

The generalization to the three-dimensional case is straightforward. In this
case, the dipole moment $\textbf{p}\coloneqq q\vec{\varepsilon}$, where
$\vec{\varepsilon}$ is the displacement vector that connects the negative to
the positive charge. Once again we take the limit $|\vec{\varepsilon}|\to 0$.
The way we have taken $q$ above, we guaranteed that the dipole moment was kept
constant, even when the charges are close enough. With this premise, the
charge distribution will be given by

$\left\langle\rho_{\varepsilon},\varphi\right\rangle\to-\frac{\partial}{\partial\textbf{p}}\varphi(0)\;,\qquad\forall\;\varphi\in\mathcal{D}(\mathds{R}^{3}),$

that we denote by

$\rho_{dip}=-\textbf{p}\cdot\nabla\delta\in\mathcal{D}(\mathds{R}^{3}).$

We now return to the main problem we intend to treat, namely, the divergence
of the electron self-energy. In this context, the fact that the charge is
fully concentrated in the origin is seen by the application of $q\delta$ on
different test functions $\varphi\in\mathcal{D}(\mathds{R}^{3})$. For any test
function whose support does not contain the origin, we have

$\int_{\textit{supp }\varphi}\rho(x)\varphi(x)d^{3}x=0.$ (16)

At the same time, the distribution possesses a finite charge once the
integration of $\rho$ over $\mathds{R}^{3}$ is equivalent to applying
$q\delta$ on a test function $\varphi\in\mathcal{D}(\mathds{R}^{3})$ whose
support contains the origin such that $\varphi(0)=1$.

$\int_{\mathds{R}^{3}}\rho(x)\varphi(x)\;d^{3}x=\int_{\textit{supp
}\varphi}\rho(x)\varphi(x)\;d^{3}x=\left\langle
q\delta,\varphi\right\rangle=q.$ (17)

This charge distribution produces both a potential $V$ and an electric field
E, which are also new distributions. We haven’t considered vector fields as
distributions so far, just like E, but this generalization is quite natural
and E acts on an element of $\mathcal{D}(\mathds{R}^{3})$ according to

$\left\langle\textbf{E},\varphi\right\rangle=(\left\langle
E_{x},\varphi\right\rangle,\left\langle
E_{y},\varphi\right\rangle,\left\langle E_{z},\varphi\right\rangle).$

The application of vector fields as distributions were mentioned here just for
completeness. It will be no longer necessary hence forth.

We can see that, in fact, $V$ represents an element of
$\mathcal{D}^{\prime}(\mathds{R}^{3})$. The explicit formula for the potential
is given by $V(\textbf{r})=\frac{e}{r}$, where $e$ is the electron charge and
we are using unities in which $4\pi\epsilon_{0}=1$, with no further
implications to the final results whatsoever. As usual, $r$ represents the
radial coordinate of a spherical coordinate system centered on the charge.
$V(\cdot)$ is a smooth function for any $\textbf{r}\neq 0$. Hence, we just
have to be concerned to the convergence of $\left\langle
V,\varphi\right\rangle$, for an arbitrary
$\varphi\in\mathcal{D}(\mathds{R}^{3})$. In effect, if $R>0$ is such that
$K=B_{R}(0)\supset\textit{supp }\varphi$ and
$M=\max_{x\in\mathds{R}^{3}}{\varphi(x)}$, then

$\begin{split}\left|\int_{\mathds{R}^{3}}V(x)\varphi(x)d^{3}x\right|&\leq
M\int_{K}V(x)d^{3}x\\\ &=M\left(\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\hskip
2.0pt\mathrm{sen}{\theta}d\theta\right)\int_{0}^{R}\frac{e}{r}r^{2}dr\\\
&=(2\pi Me)R^{2}<\infty.\end{split}$ (18)

An analogous consideration may be done for the electric field, since
$\textbf{E}\sim\frac{1}{r^{2}}$. In view of that, the radial integral will
converge as well. We can thus turn our attention to the stored self-energy of
a charged system. As we have already seen, for the particular case of an
electron there is a divergence at the origin. We are considering here only the
static case, so we don’t have to worry about magnetic fields. We can consider,
however, other cases where such divergence does not appear and we are thus
able to calculate the system self-energy. If we consider, for example, the
electron as a uniformly charged spherical shell of radius $a$, then

$\textbf{E}(\textbf{r})=\begin{cases}0&\mbox{, se }r\leq a,\\\
(e/r^{2})\;\hat{\textbf{r}}&\mbox{, se }r>a.\\\ \end{cases}$ (19)

Therefore, the self-energy, see eq. (2), will be given by

$W=\frac{1}{8\pi}\int_{\mathds{R}^{3}}\textbf{E}^{2}d\tau=\frac{1}{8\pi}(4\pi)\int_{a}^{\infty}\frac{e^{2}}{r^{2}}dr=\frac{1}{2}\frac{e^{2}}{a}.$
(20)

In distribution parlance, the last equation is just the application of the
regular distribution $\frac{1}{8\pi}\textbf{E}^{2}$ over the function
$\varphi(x)\equiv 1$. We point out that, even with
$\varphi\notin\mathcal{D}(\mathds{R}^{3})$,
$\left\langle\textbf{E}^{2},\varphi\right\rangle$ does exist. It is a
consequence of the behavior of $\textbf{E}^{2}$ at infinity, which is good
enough that we do not need to restrict the range of integration to a compact
set. For general distributions, with unknown behavior at infinity, this
restriction is considered by supposing that $\varphi$ is a test function.

We have already mentioned, see Sec. 2.2, that it is often advantageous (or
even necessary) to work with functions that are not compactly supported. This
is allowed once our distribution possesses the necessary conditions so that
its application over this larger class of functions is well behaved. For
instance, the Dirac delta may be applied on any function $\varphi$ continuous
at the origin. In our specific case, we will see how the electron self-energy
($\sim\textbf{E}^{2}$) behavior far from the origin permits such loosening of
the conditions over $\varphi$. More precisely,

$\textbf{E}^{2}=\frac{e^{2}}{r^{4}},\quad r>0$ (21)

is not well defined as a distribution. In effect, for any test function
obeying $\varphi(x)=1$ for $x$ in a neighborhood $\mathcal{V}$ of the origin,
say, a ball, we have

$\left\langle\textbf{E}^{2},\varphi\right\rangle=\int_{\mathcal{V}}\frac{e^{2}}{r^{4}}d^{3}x+\int_{\mathds{R}^{3}\backslash\mathcal{V}}\frac{e^{2}}{r^{4}}\varphi(x)d^{3}x=+\infty,$

once the first term is linear divergent due to the fourth-order homogeneity of
$\textbf{E}^{2}$.121212The classification as a linear divergence may be
justified in polar coordinates. Taking $R=1/r$, we find
$\int_{0}^{+\infty}\frac{dr}{r^{2}}=\int_{0}^{+\infty}dR$. See [3] for
details.

Outside the origin, however, $\textbf{E}^{2}$ is a smooth function, and as
such, locally integrable. Hence, we have, at least,
$\textbf{E}^{2}\in\mathcal{D}^{\prime}(\mathds{R}^{3}\backslash\\{0\\})$. For
this reason, we may extend (renormalize) the distribution $\textbf{E}^{2}$ to
a new distribution $U\in\mathcal{D}^{\prime}(\mathds{R}^{n})$ with the methods
exposed previously in Sec. 2. In that way, we expect that the electron self-
energy $E_{0}$ will be well defined as the application of $\frac{1}{8\pi}U$
over the function $\varphi\equiv 1$,

$E_{0}=\frac{1}{8\pi}\left\langle U,1\right\rangle.$ (22)

According to what we have made so far, let us first determine the scaling
degree and the singular order of $\textbf{E}^{2}$, which are key to Theorems
2.1 and 2.2. We observe that it is a homogeneous function of order $-4$, so
that, as expressed through the Example 2.1,

$\lambda^{s}\textbf{E}^{2}(\lambda x)=\lambda^{s}\frac{e^{2}}{(\lambda
r)^{4}}=\lambda^{s-4}\textbf{E}^{2},$

which implies,

$\lambda^{s}\textbf{E}^{2}(\lambda x)\xrightarrow{\lambda\to 0^{+}}0\iff s>4.$

That is, $\sigma(\textbf{E}^{2})=4$, and also
$\omega(\textbf{E}^{2})=\sigma(\textbf{E}^{2})-n=1$.

Therefore, for a test function $w\in\mathcal{D}(\mathds{R}^{3})$ and constants
$C^{0},C^{(1,0,0)}\equiv C^{1}$, $C^{(0,1,0)}\equiv C^{2}$, $C^{(0,0,1)}\equiv
C^{3}$, we obtain, according to the Theorem 2.2, a distribution
$U\in\mathcal{D}^{\prime}(\mathds{R}^{3})$ defined by

$\begin{split}\left\langle
U,\varphi\right\rangle&=\left\langle\textbf{E}_{1}^{2},W_{(1;w)}\varphi\right\rangle+\sum_{\left|\alpha\right|\leq
1}\frac{C^{\alpha}}{\alpha!}\left(D^{\alpha}\frac{\varphi}{w}\right)(0)\\\
&=\left\langle\textbf{E}_{1}^{2},W_{(1;w)}\varphi\right\rangle+C^{0}\frac{\varphi(0)}{w(0)}+\sum_{i=1}^{3}C^{i}\partial_{x_{i}}\left(\frac{\varphi}{w}\right)(0)\end{split}$
(23)

satisfying

$\left\langle
U,\varphi\right\rangle=\left\langle\textbf{E}^{2},\varphi\right\rangle\;,\quad\forall\quad\varphi\in\mathcal{D}(\mathds{R}^{3}\backslash\\{0\\}),$
(24) $\left\langle
U,wx^{\alpha}\right\rangle=C^{\alpha}\;,\quad\left|\alpha\right|\leq 1.$ (25)

Moreover, if $w$ is chosen as an Epstein-Glaser function, we have

$\left\langle
U,\varphi\right\rangle=\left\langle\textbf{E}_{1}^{2},W_{(1,w)}\varphi\right\rangle+C^{0}\varphi(0)+\sum_{i=3}C^{i}(\partial_{x_{i}}\varphi)(0).$
(26)

Before moving on, there is a comment in order. The charge distribution of a
(charged) point particle possesses spherical symmetry. It is not only due to
the concentration of charge in a single point. In fact, a electric dipole has
also a distribution whose support is contained in the origin, although the
allegedly spherical symmetry is broken once the moment p defines a privileged
direction. The additional fact that our point particle model admits no
internal structure imposes the constraint of having no special direction.

Since we would like to preserve such symmetry when extending $\textbf{E}^{2}$,
we must choose $C^{i}=0$, for $i=1,2,3$. This is justified because the last
term in (26) does not behave like a scalar under rotations of our coordinate
system, unless the three constants vanish. We can promptly see this by writing

$\sum_{i=3}C^{i}(\partial_{x_{i}}\varphi)(0)=\textbf{C}\cdot(\nabla\varphi)(0)\;,\quad\textbf{C}=(C^{1},C^{2},C^{3}).$

Now, $\nabla\varphi$ does behave as a vector, however we cannot say the same
for C. For this reason, the only way to keep this sum inert under rotations is
to set $\textbf{C}=\textbf{0}$.

From this, we may rewrite $U$ as

$\left\langle
U,\varphi\right\rangle=\left\langle\textbf{E}_{1}^{2},W_{(1;w)}\varphi\right\rangle+C^{0}\varphi(0).$
(27)

Then, we seek, analogous to what has been done in Sec. 2.2, to relax the
conditions imposed on $w$ in the renormalization of $\textbf{E}^{2}$. Our path
will be to take a sequence of test functions $w_{M}$ that converges pointwise
to $w(x)=1\in C^{\infty}(\mathds{R}^{3})$, obtaining a well defined
distribution given by

$\left\langle
U,\varphi\right\rangle=\lim_{M\to\infty}\left\langle\textbf{E}_{1}^{2},W_{(1;w_{M})}\varphi\right\rangle+C^{0}\varphi(0).$
(28)

The last equation suggests that we will take all $w_{M}$ as Epstein-Glaser
functions. Specifically, each $w_{M}$ shall be taken as

$w_{M}(x)=\begin{cases}1,&\mbox{if }x\in B_{M}(0),\\\ \chi(r-M),&\mbox{if
}x\in B_{M+1}(0)\backslash B_{M}(0),\\\ 0,&\mbox{if }x\notin
B_{M+1}(0),\end{cases}$ (29)

where $\chi$ is a smooth radial function such that $\left|\chi(s)\right|\leq
1$, $s\in[0,1]$, $\chi(0)=1$ and $\chi(1)=0$,

Hence, we have $(D^{\alpha}w_{M})(0)=\delta_{\alpha,0}$ and
$w_{M}(x)\rightarrow 1$ for any $x\in\mathds{R}^{3}$ in the limit
$M\to\infty$. Moreover, due to the result (15), for any two naturals
$M_{1},M_{2}\in\mathds{N}$ (say, $M_{1}<M_{2}$), we have

$\left\langle\textbf{E}_{1}^{2},W_{(1;w_{M_{2}})}\varphi\right\rangle=\left\langle\textbf{E}_{1}^{2},W_{(1;w_{M_{1}})}\varphi\right\rangle+\sum_{\left|\alpha\right|\leq
1}\left\langle\textbf{E}_{1}^{2},(w_{M_{1}}(x)-w_{M_{2}}(x))\frac{x^{\alpha}}{\alpha!}\right\rangle
D^{\alpha}\varphi(0),$

wherein

$w_{M_{1}}(x)-w_{M_{2}}(x)=\begin{cases}0,&\mbox{if }x\in B_{M_{1}}(0),\\\
\chi(r-M_{1})-1,&\mbox{if }x\in B_{M_{1}+1}(0)\backslash B_{M_{1}}(0),\\\
-1,&\mbox{if }x\in B_{M_{2}}(0)\backslash B_{M_{1}+1}(0),\\\
-\chi(r-M_{2}),&\mbox{if }x\in B_{M_{2}+1}(0)\backslash B_{M_{2}}(0),\\\
0,&\mbox{if }x\notin B_{M_{2}+1}(0).\end{cases}$ (30)

That way, if we denote $(a_{M})$ the real sequence whose elements are
$\left\langle\textbf{E}_{1}^{2},W_{(1;w_{M})}\varphi\right\rangle$, then we
will show that it converges, proving that $(a_{M})$ is a Cauchy sequence.
Indeed, we have just seen that131313Since
$w_{M_{1}}(x)-w_{M_{2}}(x)\in\mathcal{D}(\mathds{R}^{n}\backslash\\{0\\})$,
the action of $\textbf{E}_{1}^{2}$ may be replaced by $\textbf{E}^{2}$.

$\begin{split}a_{M_{1}}-a_{M_{2}}&=\left\langle\textbf{E}_{1}^{2},W_{(1;w_{M_{2}})}\varphi\right\rangle-\left\langle\textbf{E}_{1}^{2},W_{(1;w_{M_{1}})}\varphi\right\rangle\\\
&=\sum_{\left|\alpha\right|\leq
1}\left\langle\textbf{E}^{2},(w_{M_{1}}(x)-w_{M_{2}}(x))\frac{x^{\alpha}}{\alpha!}\right\rangle
D^{\alpha}\varphi(0),\end{split}$ (31)

so that, if we limit this sum, then we will also limit the difference
$a_{M_{1}}-a_{M_{2}}$. Now, given $\varepsilon>0$, we take $M\in\mathds{N}$
such that $\frac{1}{M}<\frac{\varepsilon}{8\pi e^{2}}$ and
$M_{1},M_{2}\in\mathds{N}$, with $M\leq M_{1}<M_{2}$. Since $\textbf{E}^{2}$
acts on test functions whose support does not contain the origin, we can
employ the formula (21), obtaining

$\left\langle\textbf{E}^{2},(w_{M_{1}}(x)-w_{M_{2}}(x))\right\rangle=\int_{\mathds{R}^{3}}\frac{e^{2}}{r^{4}}(w_{M_{1}}(x)-w_{M_{2}}(x))d^{3}x,$
$\left\langle\textbf{E}^{2},(w_{M_{1}}(x)-w_{M_{2}}(x))x_{i}\right\rangle=\int_{\mathds{R}^{3}}\frac{e^{2}}{r^{4}}(w_{M_{1}}(x)-w_{M_{2}}(x))x_{i}d^{3}x.$

Now, once i. $w_{M_{1}}-w_{M_{2}}$ is radial, ii. has support in
$B_{M_{2}+1}(0)\backslash B_{M_{1}}(0)$ and iii. assumes values only in
$[0,1]$, we have

$\left|\left\langle\textbf{E}^{2},(w_{M_{1}}(x)-w_{M_{2}}(x))\right\rangle\right|\leq(4\pi
e^{2})\int_{M_{1}}^{M_{2}+1}\frac{1}{r^{2}}dr=(4\pi
e^{2})\left[\frac{1}{r}\right]_{M_{2}+1}^{M_{1}}\leq(8\pi
e^{2})\frac{1}{M_{1}},$

that is,

$\left|\left\langle\textbf{E}^{2},(w_{M_{1}}(x)-w_{M_{2}}(x))\right\rangle\right|\leq\frac{8\pi
e^{2}}{M}<\varepsilon.$ (32)

Meanwhile,

$\left\langle\textbf{E}^{2},(w_{M_{1}}(x)-w_{M_{2}}(x))x_{i}\right\rangle=\int_{\mathds{R}^{3}}\frac{e^{2}}{r^{4}}(w_{M_{1}}(r)-w_{M_{2}}(r))x_{i}d^{3}x=0,$
(33)

because for any $i=1,2,3$, corresponding to the three Euclidean axis $x,y,z$,
respectively, the integration in $\phi$ or in $\theta$ will vanish.141414This
is only possible because $w_{M}$ is a sequence of radial functions and as
such, the integration in $r,\phi$ and $\theta$ can be factored. In fact, for
$i=1$ and $i=2$, the integrals in $\phi$ vanish,
$\int^{2\pi}_{0}d\phi\cos\phi=\int^{2\pi}_{0}d\phi\hskip
2.0pt\mathrm{sen}\phi=0$. Now, for $i=3$, we find
$\int^{\pi}_{0}d\theta\cos\theta\hskip 2.0pt\mathrm{sen}\theta=0$.

Hence, with the help of (32) and (33), we may conclude that

$\left|a_{M_{1}}-a_{M_{2}}\right|\leq\varepsilon\varphi(0),$ (34)

which, in turn, implies that $(a_{M})$ will be a Cauchy sequence, that is, a
convergent one.

All the previous development allows us to show that the distribution defined
in (28), that we will denote simply by

$\left\langle
U,\varphi\right\rangle=\left\langle\textbf{E}_{1}^{2},W_{(1;1)}\varphi\right\rangle+C^{0}\varphi(0),$
(35)

is well defined. Thus, we may finally use the eq. (22) to obtain the
renormalized electron self-energy,

$E_{0}=\frac{1}{8\pi}\left\langle\textbf{E}_{1}^{2},W_{(1;1)}1\right\rangle+\frac{1}{8\pi}C^{0}=\frac{C^{0}}{8\pi}.$
(36)

Although simple, the eq. (36) bears great physical meaning, concentrating our
results. We have shown that defining the electron self-energy as the
application of the distribution
$\frac{1}{8\pi}U\in\mathcal{D}(\mathds{R}^{3})$, which extends (or
renormalize) $\textbf{E}^{2}$, over the constant function $1$, we get rid of
the divergence previously depicted. This divergence appeared when one directly
considers $\textbf{E}^{2}$, which cannot be seen as an actual
distribution.151515The statement that
$\textbf{E}^{2}\notin\mathcal{D}(\mathds{R}^{3})$ is a consequence that the
product between distributions is not, in general, well defined. An alternative
method to skirt the electron self-energy divergence is related to a
generalization to the very concept of distributions, working with the
generalized Coulombeau functions. For details, see [22, 23] and references
therein. Now, $E_{0}$ becomes an undetermined constant, that we may control to
serve our model. This is the very kernel of a renormalization method. If, for
instance, we assume that the self-energy is, alone, responsible for the
electron mass, we shall take

$C^{0}=8\pi m_{e}c^{2},$

in a way that $E_{0}=m_{e}c^{2}$.

To summarize, we have seen how the self-energy problem originates in the fact
that $\textbf{E}^{2}$ is not a proper element of
$\mathcal{D}(\mathds{R}^{n})$. On the other hand, $\textbf{E}^{2}$ is, outside
the origin, a smooth function. Thus, we at least have
$\textbf{E}^{2}\in\mathcal{D}^{\prime}(\mathds{R}^{n}\backslash\\{0\\})$,
which means we can use Theorem 2.2 to extend it to a distribution in
$\mathcal{D}^{\prime}(\mathds{R}^{n})$.

##  4 Conclusion

The main objective of this work was to analyze (and renormalize) a simple but
central problem in classical electrodynamics: the electron self-energy.
Although the electrostatics model of a charged point particle implies a linear
divergence on the self-energy, we may skirt this infinity with an extension of
the corresponding distribution. With more details,

1. We have shown how distributions and also their extensions can play a central role already in classical electrodynamics. Even the most basic physical quantities such as charge density, scalar and vector fields turn out to be described as linear functionals. In addition, we have also demonstrated that a specific scheme of renormalization, seen as an extension of a particular distribution, can be applied to overcome the divergent electron self-energy.

2. The leading results of extension of distributions, that is, the corresponding renormalization, were all enunciated, following the lines in [24]. We have focused in a particular subspace of the set of test functions, namely, the one whose elements vanish in an arbitrary neighborhood of the origin. We have investigated the behavior of different distributions in the origin and how one could recover such distributions, in the sense of making them continuous linear functionals defined over all the space of test functions.

3. We have applied the concepts of distributions and the corresponding extensions to the classical electron self-energy. At first sight, electrostatics implies a divergence once we treat the electron as a charged point particle. However, our construction shows that its self-energy turns out to be an undetermined constant upon renormalization, so that our parameters might be fixed, for example, appealing to empirical results.

In conclusion, although the use of distribution-valued fields (and their
renormalization as extensions) in classical electrodynamics may seem abstract
and disconnected from physical reality, it is an important mathematical tool
that enables us to describe the behavior of physical fields in a rigorous and
consistent manner. The key is to choose renormalization conditions that
correspond to physical observables. In our case, we are using the energy,
which could potentially be measured and accepted as an observable.

## References

  * [1] Zee, A.: Quantum field theory in a nutshell. Princeton University Press, Princeton (2010)
  * [2] Streater, R. F. and Wightman, A. S.: PCT, Spin and Statistics, and All That. Princeton University Press, Princeton (2001)
  * [3] Bovy, J.: The self-energy of the electron: A quintessential problem in the development of QED. Preprint at https://arxiv.org/abs/physics/0608108 (2006)
  * [4] Griffiths, D. J.: Introduction to Electrodynamics. Prentice Hall, New Jersey (1999)
  * [5] Jackson, J. D.: Classical electrodynamics. John Wiley & Sons, Inc., New Jersey (1999)
  * [6] Hudson, J. J., Kara, D. M., Smallman, I. J., Sauer, B. E., Tarbutt, M. R. and Hinds, E. A.: Improved measurement of the shape of the electron. Nature 473, 493-496 (2011)
  * [7] ACME Collaboration, Improved limit on the electric dipole moment of the electron. Nature 562, 355-360 (2018)
  * [8] Griffiths, D. J.: Introduction to elementary particles. Wiley-VCH, Weinheim (2011)
  * [9] Ryder, L. H.: Quantum Field Theory. Cambridge University Press, Cambridge - NY (1996)
  * [10] Bogoliubov, N. N. and Shirkov, D. V.: Introduction to the theory of quantized fields. John Wiley & Sons, Inc., New York (1980)
  * [11] Epstein, H. and Glaser, V.: The role of locality in perturbation theory. Annales de l’I.H.P. Physique Théorique 19, 211-295 (1973)
  * [12] Gracia-Bondía, J. M.: Improved Epstein-Glaser renormalization in coordinate space i. Euclidean framework. Mathematical Physics, Analysis and Geometry 6, 59-88 (2003)
  * [13] Gracia-Bondia, J. M. and Lazzarini,S.: Connes-Kreimer-Epstein-Glaser renormalization. Preprint at https://arxiv.org/abs/hep-th/0006106 (2000)
  * [14] Grangé, P. and Werner, E.: UV and IR behaviour for QFT and LCQFT with fields as operator valued distributions: Epstein and Glaser revisited. Nuclear Physics B - Proceedings Supplements 161, 75-80 (2006)
  * [15] Grangé, P. C. and Werner, E.: Quantum fields as operator valued distributions and causality. Preprint at https://arxiv.org/abs/math-ph/0612011v2 (2007)
  * [16] Prange, D.: Epstein-Glaser renormalization and differential renormalization. Journal of Physics A: Mathematical and General 32, 2225-2238 (1999)
  * [17] Schwartz, L.: Mathematics for the Physical Sciences. Dover Publications, New York (2008)
  * [18] Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (2003)
  * [19] Richards, I. and Youn, H.: Theory of distributions: a non-technical introduction. Cambridge University Press, Cambridge (1990)
  * [20] Schwartz, L.: Théorie des Distributions. Actualités scientifiques et industrielles, Hermann (1957)
  * [21] Lemos, N. A.: Convite à Física Matemática. Editora Livraria da Física, São Paulo (2013)
  * [22] Gsponer, A.: The classical point electron in Colombeau’s theory of nonlinear generalized functions. Journal of Mathematical Physics 49, 102901 (2008)
  * [23] Gsponer, A.: A concise introduction to Colombeau generalized functions and their applications in classical electrodynamics. European Journal of Physics 30, 109-126 (2009)
  * [24] Fredenhagen, K.: Quantum Field Theory. Preprint at https://www.physik.uni-hamburg.de/th2/ag-fredenhagen/dokumente/qft09-10.pdf (2009)