11institutetext: Dipartimento di Fisica e Astronomia, Università di Padova,
Vicolo dell’Osservatorio 3, I-35122, Padova, Italy 22institutetext: INAF -
Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122,
Padova, Italy 33institutetext: Instituto de Física y Astronomía, Universidad
de Valparaíso, Avda. Gran Bretaña 1111, Valparaíso, Chile 44institutetext:
Université de Strasbourg, CNRS, Observatoire astronomique 392 de Strasbourg,
UMR 7550, 67000 Strasbourg 55institutetext: Aix Marseille Université, CNRS,
CNES, LAM, Marseille, France 66institutetext: Observatoire de Genève,
Université de Genève, 51 Ch. des Maillettes, 1290 Versoix, Switzerland
77institutetext: IPAC, California Institute of Technology, 1200 East
California Boulevard, Pasadena, CA 91125, USA 88institutetext: Department of
Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford
OX1 3RH, UK 99institutetext: INAF - Osservatorio di Astrofisica e Scienza
dello Spazio, Via Gobetti 93/3, I-40129, Bologna, Italy 1010institutetext:
Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica,
Chile 1111institutetext: University of Wisconsin, 475 N Charter Str.,
Madison., WI, USA 1212institutetext: University of Massachusetts Amherst, 710
North Pleasant Street, Amherst, MA 01003-9305, USA 1313institutetext:
Dipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1,
50019, Sesto Fiorentino (Firenze), Italy 1414institutetext: INAF -
Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125, Firenze, Italy
1515institutetext: Instituto de Física, Pontificia Universidad Católica de
Valparaíso, Casilla 4059, Valparaíso, Chile 1616institutetext: Center for
Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY
10010, USA 1717institutetext: Space Telescope Science Institute, 3700 San
Martin Dr., Baltimore, MD 21218, USA 1818institutetext: Department of Physics
and Astronomy, University of California Davis, One Shields Avenue, Davis, CA
95616, USA 1919institutetext: Gemini Observatory, NSF’s NOIRLab, 670 N.
A’ohoku Place, Hilo, Hawai’i, 96720, USA 2020institutetext: Cosmic Dawn Center
(DAWN), Jagtvej 128, DK2200 Copenhagen N, Denmark 2121institutetext: DTU-
Space, Technical University of Denmark, Elektrovej 327, DK2800 Kgs. Lyngby,
Denmark 2222institutetext: Niels Bohr Institute, University of Copenhagen,
Jagtvej 128, DK-2200 Copenhagen N, Denmark 2323institutetext: Departamento de
Astronomía, Universidad de La Serena 2424institutetext: Instituto de
Investigación Multidisciplinar en Ciencia y Tecnología, Universidad de La
Serena 2525institutetext: Department of Space, Earth and Environment, Chalmers
University of Technology, Onsala Space Observatory, 439 92 Onsala, Sweden
2626institutetext: Dipartimento di Fisica e Astronomia, Università di Bologna,
via Gobetti 93/2, 40129, Bologna, Italy 2727institutetext: National Centre for
Nuclear Research, ul. Pasteura 7, 02-093, Warsaw, Poland

# The ALMA-ALPINE [CII] survey: sub-kpc morphology of 3 main-sequence galaxy
systems at z$\sim$4.5 revealed by ALMA

T. Devereaux<EMAIL_ADDRESS>P. Cassata 1122 E. Ibar 33 C.
Accard 44 C. Guillaume 44 M. Béthermin 4455 M. Dessauges-Zavadsky 66 A. Faisst
77 G. C. Jones 88 A. Zanella 22 S. Bardelli 99 M. Boquien 1010 E. D’Onghia
1111 M. Giavalisco 1212 M. Ginolfi 13131414 R. Gobat 1515 C. C. Hayward 1616
A. M. Koekemoer 1717 B. Lemaux 18181919 G. Magdis 202021212222 H. Mendez-
Hernandez 23232424 J. Molina 2525 F. Pozzi 262699 M. Romano 272722 L. Tasca 55
D. Vergani 99 G. Zamorani 99 E. Zucca 99

(Received XXX; accepted XXX)

###### Abstract

Context. From redshift 6 to redshift $\approx$ 4 galaxies grow rapidly from
low mass galaxies towards the more mature massive galaxies we see at the
cosmic noon. Growth via gas accretion and mergers undoubtedly shape this
evolution - however, there currently exists much uncertainty over the
contribution of each of these processes to the overall evolution of galaxies.
Furthermore, previous characterisations of the morphology of galaxies in the
molecular gas phase has been limited by the coarse resolution of previous
observations.

Aims. The goal of this paper is to derive the morpho-kinematic properties of 3
main-sequence systems at $z\sim 4.5$, drawn from the ALPINE survey, using
brand new high-resolution ALMA data in band 7. The objects were previously
characterised as one merger with three components, and and two dispersion-
dominated galaxies.

Methods. we use intensity and velocity maps, position-velocity diagrams and
radial profiles of [CII], in combination with dust continuum maps, to analyse
the morphology and kinematics of the 3 systems.

Results. In general, we find that the high-resolution ALMA data reveal more
complex morpho-kinematic properties. We identify in one galaxy interaction-
induced clumps, showing the profound effect that mergers have on the molecular
gas in galaxies, consistent with what is suggested in recent simulations. A
galaxy that was previously classified as dispersion dominated turned out to
show two bright [CII] emission regions, that could either be merging galaxies
or massive star-forming regions within the galaxy itself. The high resolution
data for the other dispersion dominated object also revealed clumps of [CII]
that were not previously identified. Within the sample, we might also detect
star-formation powered outflows (or outflows from Active Galactic Nuclei)
which appear to be fuelling diffuse gas regions and enriching the
circumgalactic medium.

###### Key Words.:

Galaxies:evolution-Galaxies:high-redshift–Galaxies:ISM–Galaxies:star
formation–Submillimeter:galaxies

##  1 Introduction

The epoch between z$\sim 6$ and $z\sim 2$ is crucial for understanding galaxy
evolution: during these $\sim$2 Gyr of cosmic time the star-formation rate
density of the Universe (SFRD) increases by a factor of 20 (Madau & Dickinson,
2014), with the average specific star-formation rate of individual galaxies
going up by about the same amount (Tasca et al., 2015; Schreiber et al., 2015;
Tomczak et al., 2016); over the same the same timespan, galaxies evolve from
small, metal poor, dust free, and turbulent objects in the very Early Universe
(Maiolino et al., 2023; Finkelstein et al., 2022) to more regular morphologies
at z$\sim$2 (van der Wel et al., 2014; Zhang et al., 2019), with ordered disks
(Förster Schreiber et al., 2009), and significant dust and metal content
(Troncoso et al., 2014; Popping et al., 2023).

While it is thought that galaxies at these epochs grow via mergers and the
accretion of gas from galactic halos, the relative importance of the two
mechanisms is not clear: (Förster Schreiber & Wuyts, 2020) suggest that gas
accretion dominates, while (Romano et al., 2021a) in addition to simulations
(i.e. Vallini et al., 2013; Kohandel et al., 2019; Pallottini et al., 2022a),
suggest a build up from a combination of major mergers, minor mergers and gas
accretion. In order to make significant progress, it is vital to unveil the
morpho-kinematic properties of galaxies $4<z<6$, when these mechanisms are
working at full strength, at the same time revealing the interplay between
stars, dust and molecular gas, that constitutes the fuel from which stars can
form.

While a combination of ground based telescope. such as the Visible and
Infrared Survey Telescope for Astronomy (VISTA) with space observatories such
as Hubble Space Telescope (HST), Spitzer, and James Webb Space Telescope
(JWST) can probe stellar component of galaxies up to z$\sim$6 and beyond (Song
et al., 2016; Naidu et al., 2022; Harikane et al., 2023), observing molecular
gas at redshifts beyond the peak epoch of cosmic star formation is
challenging. H2 cannot be observed directly, so studies have used tracers such
as CO (see review by Bolatto et al., 2013) and [CI] (e.g., Walter et al.,
2011). However, the low excitation transitions of the CO molecule ((1-0) and
(2-1)), that better probe the molecular gas content of galaxies, are
inaccessible to the Atacama Large Molecular Array (ALMA) beyond z$\sim$3
Carilli & Walter (2013), and are too faint to be detected with reasonable
integration times with Very Large Array (VLA) (D’Eugenio et al., 2023). This
has led in recent years to use the [CII]$\lambda$158$\mu$m transition as a
tracer for the molecular gas at high redshifts. The [CII] emission line is one
of the brightest lines in the spectra of galaxies at high redshifts - owing to
its ability to trace the ionised, molecular and neutral phases. Whilst [CII]
has typically been used as a star formation tracer (e.g., De Looze et al.,
2014), more recent work suggests that the majority of the [CII] flux in
galaxies comes from molecular clouds. Calibrations of the [CII] line as a
molecular gas tracer in high redshift galaxies have been studied in Zanella et
al. (2018). In Vizgan et al. (2022) it is was later confirmed that the [CII]
luminosity is more tightly related to the $\rm-M(H_{2})$, with the correlation
between [CII] and SFR arising from the Kennicutt-Schmidt relation (Kennicutt,
1998).

The ALMA Large Program to INvestigate [CII] at Early times (ALPINE) survey (Le
Fèvre et al., 2020; Béthermin et al., 2020; Faisst et al., 2020) was designed
exactly to explore the interplay between stars, dust and molecular gas mass in
normal main-sequence galaxies at $4.5<z<6$ \- the redshift range where
galaxies transition from lower mass primordial galaxies to more evolved
galaxies at cosmic noon. [CII] was used as a tracer for molecular gas to
reveal the molecular gas content of galaxies (Dessauges-Zavadsky et al.,
2020): using the calibration provided in Zanella et al. (2018), the authors
found a good agreement between [CII] luminosity, the dust continuum
(extrapolated from the continuum flux) and dynamical masses, supporting [CII]
being a reliable tracer of molecular gas. Le Fèvre et al. (2020) and (Romano
et al., 2021a) (see also Jones et al., 2021) found that 40% of galaxies in the
sample of 118 galaxies are mergers, 20% were found to be characterised by a
high velocity dispersion and extended components, and 11% of galaxies were
found to be rotating disks.

Among the ALPINE galaxies that are resolved, a large number appear to have an
underlying gas halo (Fujimoto et al., 2020). Recent semi-analytical models
indicate these halos can be produced by star-formation driven outflows
(Pizzati et al., 2022). Due to the limited ($\sim 1"$) resolution of ALPINE,
the majority of galaxies were poorly resolved, meaning diffuse gas could not
be distinguished from compact regions. Furthermore, The low resolution
severely impacted the ability of ALPINE to trace the properties (i.e. mergers,
clumps and components in velocity) in (some of) these galaxies.

Simons et al. (2019) shows that at low resolution close mergers can be
indistinguishable from disks. This is an issue with the majority off galaxies
observed in [CII] thus far. The low resolution ($\sim 0.7\arcsec$) of the ALMA
observations for the ALPINE survey has limited the number of systems of
multiple galaxies which can be spatially resolved from one another and thus
can be identified as mergers. One of the best examples of a merger in [CII] is
extensively studied in Jones et al. (2020). The authors used ALPINE data to
show a massive system undergoing mass assembly via the merger of 3 galaxies.
This example possibly presents the most idealised case of a major merger in
which 3 merging galaxies can be clearly resolved from each other spatially as
well as in velocity.

When resolved at kpc and sub-kpc scales, galaxies can appear clumpy in star
formation tracers (e.g., Claeyssens et al., 2023): the coarse resolution of
ALPINE, allowing to resolve scales of $\sim 5-8$ kpc, is not therefore
sufficient to resolve these clumps spatially. The predominant theory is that
clumpiness comes from gravitational instability (Dekel et al., 2009) or via
mergers/interactions (Di Matteo et al., 2008). At this redshift, with the
resolution of current observations, it is suggested we are mainly observing
mergers Zanella et al. (2021). The lack of high resolution spectroscopic data
sets has significantly hindered our ability to uncover the kinematics in these
galaxies and detect the imprints of mergers upon galaxies and subsequently how
they evolve.

There exist only a restricted number of spatially resolved observations,
targeting molecular gas, in the high redshift ($z>4$) regime. Roman-Oliveira
et al. (2023) studied 4 starburst galaxies, finding the majority of them to be
rotating disks. Observations of merging systems (Oteo et al., 2018; Litke et
al., 2019) identified irregular components of gas emission. However, these
surveys all traced star-bursting galaxies with $\rm
SFRs>1000\,{M_{\odot}\,yr^{-1}}$, providing insights into only a subdominant
population of galaxies at this redshift. Herrera-Camus et al. (2021a) traced a
z$\sim$5.5 typical star forming galaxy at $0.3\arcsec$ angular resolution
finding evidence of star formation outflows, extended emission and as well as
rotation. The authors also found evidence of an extended molecular gas halo
surrounding the galaxy. The small number of spatially resolved galaxies at
this redshift leaves much uncertainty for the processes occurring in these
galaxies - particularly for main sequence mergers/dispersion dominated
galaxies (which were found to be the majority from ALPINE).

In this paper we utilise brand new high resolution ALMA [CII] observations to
analyse 3 main sequence galaxy systems at resolutions of up to $0.15\arcsec$
and investigate the morphology and kinematics on a kpc scale and understand
the processes at play and the classifications of galaxies at high resolution.
Giving one of the best insights into the molecular gas of main sequence of
galaxies undergoing mass assembly.

This paper assumes a $\Lambda$-CDM cosmology, with $h_{0}$=0.7,
$\Omega_{\Lambda}=0.3$ and $\Omega_{m}=0.3$.

##  2 The Sample studied in this paper

This paper focuses on 3 galaxies at redshifts $\sim$4.5 that were studied in
Jones et al. (2021). All were found to be on the ’main sequence’ at this
redshift (Faisst et al., 2020). These galaxies were selected for high-
resolution follow-up as they were: i) the 3 brightest galaxies observed in
[CII] in the ALPINE survey, ii) they had a strongly detected continuum
(¿7-$\sigma$) iii) the [CII] and continuum appeared aligned in ALPINE
observation, and iv) finally the HST images of these galaxies showed no
evidence of late stage mergers. In this section, we present the properties of
these galaxies, as obtained from the analysis of the low resolution ALPINE
data.

###  2.1 DC8737

DEIMOS_COSMOS_873756 hereafter DC8737 is the brightest galaxy in the ALPINE
survey and is at a spectroscopic redshift of $z=4.5480$. At the coarse $\sim
0.7$ resolution of the original ALPINE data, it shows no signs of rotation but
a high velocity dispersion. The [CII] emission is extended to the East. The
galaxy was classified as dispersion dominated due to the presence of multiple
components visible in data obtained with the Subaru telescope - as well as the
aforementioned extended, fainter [CII] substructures Jones et al. (2021). In
Cassata et al. (2020) the galaxy is found to be a Ly$\alpha$ emitter in the
central region.

###  2.2 VC8326

VUDS_COSMOS_5101218326 (hereafter VC8326) has a redshift $z=4.5678$. Jones et
al. (2021) modelled its velocity map with 3D-Based Analysis of Rotating Object
via Line Observations (3D-BAROLO) Di Teodoro & Fraternali (2015), finding that
this object is likely to be an extended dispersion-dominated galaxy. HST/ACS
F814W imaging (Koekemoer et al., 2007) from the COSMOS survey, targetting the
rest-frame UV, reveal extended emission over 1” scales. VC8326 is in the
central regions of the most massive component of the PCl J1001+0220
protocluster (Lemaux et al. (2018), Staab et al. submitted).

###  2.3 DC8187

DEIMOS_COSMOS_818760 (hereafter DC 8187) is constituted of 3 galaxies at
$z=4.56038$, $z=4.56628$ and $z=4.56229$, extensively studied in Jones et al.
(2020). The eastern and central sources are the 2 largest and brightest and
are found to be close both spatially and in velocity, and therefore are
considered to be undergoing a major merger. A third source (separate in
velocity by $\sim 300\,{\rm km\,s^{-1}}$ and spatially by $18\,{\rm kpc}$) is
detected nearby. It is likely going to merge with the main system in the
future. The velocity map of this galaxy showed signs of rotation or tidal
disruption and it was found in Jones et al. (2020) to have the lowest gas mass
of the 3 sources.

##  3 Data Processing and Analysis

###  3.1 Observations and Data reduction

The data used in this paper were obtained between 2020 and 2021 as part of the
ALMA project 2019.1.00226.S (PI E. Ibar). Configurations C43-5 and C43-3 were
used, producing resolutions of $0.15\arcsec$ (high resolution, HR) and
$0.3\arcsec$ (medium resolution, MR), respectively. Objects were observed for
937 s and 1975 s in the C43-5 and C43-3, respectively, only a fraction of the
30 h originally planned for the observations. The spectral setup was such that
all three galaxies were able to be observed within a loop with exactly the
same spectral setup, using the same phase calibrator - this was to reduce the
overheads and maximize the total integration times. Additionally, we also
included in the analysis the lower resolution (LR) data - available from
ALPINE (2017.1.00428.L, PI Le Fèvre).

The High resolution, Medium resolution, and Low resolution data were
processed, independently for each observation, using the standard ALMA
pipeline to produce calibrated the uv datasets, for each observation, for each
source. The pipeline calibration was manually checked and no issues were found
with the calibration. The line emission was conservatively estimated and the
continuum subtracted using the CASA task uvcontsub to produce the continuum
subtracted visibility file for each resolution.

Due to the nature of interferometry, high resolution images typically fail to
pick up extended emission (without long integration times - which are
unrealistic in the case of a competitive interferometer such as ALMA).
However, it is possible to concatenate low resolution data with higher
resolution data to build up a picture of both the compact and extended
emission. Therefore, a combination of visibility files is used in this paper.
First, a concatenation of the High resolution and Medium Resolution was made
using concat for each source. It was decided to combine the highest
resolutions observations ($0.15\arcsec$ and $0.3\arcsec$) to a single
concatenation in order to maximise the S/N. Secondly, a High resolution +
Medium resolution + Low resolution concatenation was made using the CASA task
concat for each source. This concatenation allows for a better coverage of the
uv-plane, allowing the tracing of both extended and compact components of
emission. From here on out, we will refer to these concatenations as
High+Medium+Low resolution (HR+MR+LR) - for all 3 resolutions combined and
HR+MR - for the high and medium resolution concatenation.

In order to produce the final data cubes, cleaning was undertaken using the
CASA task tclean, a Briggs weighting (with robust = 0.5) was chosen to best
compromise between resolution and sensitivity. A dirty image was produced, and
the RMS per pixel, of the peak channel, estimated using an aperture located
outside of the galaxy (to avoid emission from the galaxy). All galaxies were
cleaned to a threshold of $2.5\times{\rm RMS}$. The resultant RMS per pixel
and synthesised beam for each moment 0 map are shown in Table LABEL:Table_1
and Table LABEL:Beam.

Table 1: The RMS of each [CII] intensity map (moment 0) in Jy km s-1 beam-1 Galaxy | HR+MR | HR+MR+LR
---|---|---
VC8326 | 64 | 70
DC8737 | 76 | 59
DC8187 | 78 | 67
Table 2: The beam size of each resolution concatenation in arc seconds. These are the best-fit 2D Gaussian FWHMs of the synthesised beam Galaxy | HR+MR Beam | HR+MR+LR Beam
---|---|---
VC8326 | $0.232\times 0.158$ | $0.313\times 0.242$
DC8737 | $0.224\times 0.154$ | $0.349\times 0.273$
DC8187 | $0.241\times 0.179$ | $0.294\times 0.231$

###  3.2 Production of moment 0 and 1 maps

For each source, at each resolution, moment 0 and moment 1 maps were produced
for channels which contain line emission. The moment 0 map shows the spatial
distribution of line emission, whereas the moment 1 map shows the velocity
distribution of line emission. The optimum velocity range of the 1D spectra
which maximises the S/N ratio of the line emission was found using the
procedure used in Zanella et al. (2018), this is described as follows. The
total signal to noise (S/N) was measured within an aperture of diameter
$3\arcsec$ (for the HR+MR+LR resolution concatenation) or $1.5\arcsec$ for the
HR+MR cubes. This was calculated for all possible channel ranges in the data-
cube, the velocity range which maximised the S/N is considered to be the
channel range containing line emission.

Moment 0 and 1 maps were produced using the CASA task immoments using the
channels previously selected to contain line emission. For the HR+MR and
HR+MR+LR moment 1 maps,a $\sigma$-cut was made (between 3-5$\sigma$ depending
on the galaxies morphology), i.e. any pixel with a S/N less than defined above
is set to null. This ensures the velocity map is only showing pixels likely to
be a constituent of the galaxy.

Continuum maps were produced for each source. These were produced directly
from non-continuum subtracted visibility files. The same concatenations as
described in Sect.3.1 were created. All spectral windows in the band were
selected, however the channels found to contain line emission - described
above - were excluded. Using tclean, with a multifrequency mode and a Briggs
weighting (Robust = 0.5), a dirty continuum image was produced for each
observation. Using an aperture of an empty background region, the RMS was
estimated and cleaning was repeated to 2.5$\times{\rm RMS}$ in order to
produce clean continuum maps.

The moment 0 maps and moment 1 maps are shown in Figures 1, 2, and 3. For the
HR+MR+LR maps, which give the best view of the compact + extended emission, K
band, Visible and Infrared Survey Telescope for Astronomy (VISTA) observations
(matching the rest-frame optical) and HST observations (matching the rest-
frame ultraviolet) are contoured - in order to compare the stellar
distribution and obscured star formation to the [CII] distribution. The
Continuum maps are also shown in the same Figures - along with [CII] contours
to compare the obscured star formation (from continuum) and gas distributions
(shown by the [CII]).

Figure 1: Summary of maps produced for for DC8737. The top line shows the
moment 0 maps, from left to right: HR+MR (with S/N contours from $\pm
5,6,7,\cdots\sigma$), HR+MR+LR - with S/N contours of $\pm
5,6,7,\cdots\sigma$, as well as HST/ACS F814W (Koekemoer et al. 2007; blue;
3,5,7,9 $\sigma$) and K band (red; 3,4 $\sigma$), continuum map (with
continuum and [CII] contours from $5\sigma$ in purple and black,
respectively). The second line shows the moment 1 maps cut to 5 $\sigma$, for
the HR+MR and the HR+MR+LR configurations, and the PV diagram taken in the
HR+MR+LR concatenation - using the slit shown on the top line - with contours
of 0.7, 0.8, and 0.9x the peak value.) Figure 2: Summary of maps produced for
for VC8326. The top line shows the moment 0 maps of VC8326, from left to
right: HR+MR (with contours from $\pm 3,4,5,\cdots\sigma$), HR+MR+MR - with
contours from $\pm 3,4,5,\cdots\sigma$\- as well as HST/ACS F814W (Koekemoer
et al. 2007; blue; 3,5,7,9 $\sigma$) and K band (red; 3, 13, 23, 33 $\sigma$),
continuum map (with continuum and [CII] contours from $3\sigma$ in purple and
black, respectively). The second line shows PV diagram (from the slit shown on
the top line taken in the HR+MR concatenation) with contours of 0.7, 0.8, and
0.9x the peak value) and the moment 1 maps which are cut to 3 and 5 $\sigma$,
respectively. Figure 3: Summary of maps produced for for DC8187. The top row
shows the moment 0 maps, from left to right: HR+MR (with contours from $\pm
5,6,7,\cdots\sigma$), HR+MR+LR (with contours from $\pm 5,6,7,\cdots\sigma$).
- as well as HST/ACS F814W (Koekemoer et al. 2007; blue; 3,5,7,9 $\sigma$) and
K band (red; 3,4,5,6 $\sigma$), continuum map (with continuum and [CII]
contours from $5\sigma$ in purple and black, respectively). The second line
shows the moment 1 maps shown in the HR+MR concatenation and are cut to 5
$\sigma$. On the right the second line shows the PV diagram,taken in an East-
West slit in the HR+MR concatenation with contours of 0.7, 0.8, and 0.9x the
peak value.

###  3.3 Fidelity of Galaxy Components

The moment 0 maps show that the high resolution observations reveal complex
components (i.e. clumps, compact regions) that were previously not observable
due to the coarse resolution of the ALPINE survey. However, there is a risk
that spurious emission from noise can be misidentified as emission from the
galaxy. Therefore, to ensure any detections are significant, we adopt the
method from Walter et al. (2016): assuming that the noise is non-Gaussian, the
number of positive and negative peaks, above a given flux limit, should be the
same, if no detection is present. Real and robust detections should instead
only appear as positive peaks. Using equation 1 we estimate the term coined in
Walter et al. (2016) ’fidelity’.

$Fidelity(S/N)=1-\frac{N_{neg}(S/N)}{N_{pos}(S/N)}$ (1)

In practice, the fidelity compares the ratio of non physical negative noise
peaks to potentially real positive peaks in signal for a given signal to noise
ratio. The S/N where this fidelity is equal to 60% is the value that is
considered the minimum for a detection to be considered real, for a given
number of connected pixels. In order to estimate the fidelity of individual
components, we count the number of positive sources of comparable or larger
number of connected pixels as the component at a given S/N, and the same is
conducted for negative sources; the comparison of the two, according to
equation 1, gives the fidelity of such component.

###  3.4 [CII] flux analysis

Performing the photometry of the [CII] and continuum emission in these
galaxies is a non trivial task. DC8737, DC8187 and VC8326 have clumpy or
irregular morphologies which are poorly fit by a two dimensional Gaussian.
Therefore, aperture photometry was conducted on the moment 0 maps, with
aperture sizes that must be carefully selected as not to miss extended faint
emission, and at the same time minimising the noise (which increases with
aperture size). The noise was estimated with a consistent method to that used
in the ALPINE survey (Béthermin et al., 2020) by scaling the error associated
to a single beam with the number of synthesised beams per aperture. We note
that we did not include flux calibration errors - which can amount to 10% of
the total flux.

In the HR+MR+LR concatenation, apertures were selected from diameters smaller
than $3\arcsec$ in diameter for all sources, which this is consistent with the
size chosen for the original ALPINE sources in Béthermin et al. (2020). To
minimise the effect of noise (which increases with larger aperture sizes),
while preventing emission from being missed. The aperture radius is selected
by finding the radius where the S/N is maximised. Beyond this point, the
additional flux (from an increased aperture) becomes less significant than the
increased error (from noise) - in order to minimise the error whilst ensuring
no flux is lost. For the HR+MR concatenation the apertures are generally
significantly smaller than the HR+MR+LR concatenation due to fact they trace
preferentially the compact emission (and have higher RMS values). We emphasise
that when the same aperture is used as in the HR+MR+LR concatenation the
HR+MR+LR flux is recovered (within the large errors).

For flux measurements of features within galaxies (clumps or other components)
the apertures were manually adjusted, as they are poorly fit by a 2d Gaussian.
In these cases, we made sure to ensure that the aperture size was larger than
the synthesised beam to ensure an accurate measurement - by preventing flux
being lost.

Table 3: Summary of integrated [CII] intensities for each source. Galaxy | MR+HR flux | Aperture Radius | LR+MR+HR flux | Aperture radius | LR ALPINE flux
---|---|---|---|---|---
|  Jy km s-1 | kpc |  Jy km s-1 | kpc |  Jy km s-1
DC8737 | $4.32\pm 0.27$ | 2.5 | $6.8\pm 0.26$ | 4.5 | $5.84\pm 0.27$
VC8326 | $2.89\pm 0.36$ | 4.0 | $3.31\pm 0.29$ | 4.0 | $2.92\pm 0.23$
DC8187(E) | $3.05\pm 0.19$ | 2.9 | $3.54\pm 0.23$ | 4 | $2.6\pm 0.20$
DC8187(C) | $4.37\pm 0.19$ | 2.25 | $5.3\pm 0.23$ | 4.4 | $4.9\pm 0.30$
DC8187(W) | $0.74\pm 0.15$ | 1.0 | $0.95\pm 0.19$ | 1.4 | $0.8\pm 0.20$

The integrated fluxes measured for each source are reported in Table
LABEL:Table_2. For DC8187, photometric measurements were conducted
individually for each constituent of the merging system (whilst ensuring no
overlap between the apertures). It is clear that the [CII] flux measured for
the objects varies significantly across different concatenations, as described
in detail in the following subsections.

####  3.4.1 DC8737

DC8737 is the galaxy with the brightest [CII] in the ALPINE survey, and shows
the largest discrepancy between [CII] fluxes measured in the different
configurations. The [CII] emission appears to come from a single component, in
all configurations (see the first 2 lines of Figure1).

The total emission in the HR+MR+LR concatenation is 6.80 $\pm$ 0.26 Jy km s-1,
60% larger than the flux of the HR+MR concatenation (4.32 $\pm$ 0.27 Jy km
s-1). This indicates that this object has a significant extended component,
indicative of diffuse gas, which is not recovered by the configurations with
the highest resolutions. The ALPINE flux is even lower, at 5.84 $\pm$ 0.27 Jy
km s-1 \- whilst this is lower than the flux recovered in this paper - it
still supports the presence of an extended gas component.

####  3.4.2 VC8326

The [CII] emission in VC8326 is resolved into 2 compact regions when the HR
and MR resolution data are included, as can be seen in line 1 of Figure 2.
These compact regions could be interpreted as two merging galaxies (or
clumps).

Differently from DC8737, the [CII] fluxes measured for VC8326 in the different
configurations all agree within the errors: the fluxes are 2.89, and 3.31 Jy
km s-1, for the MR+HR, and LR+MR+HR, respectively. The total recovered fluxes
in these new concatenations are, within uncertainties, comparable to the
original ALPINE flux.

In addition, we estimated, in the HR+MR+LR concatenation, the amount of [CII]
emission that can be associated to the two main components of the objects
identified in Figure 2. We report in Table LABEL:Table_3 the apertures that we
used and the results of this analysis. We find that the 2 compact regions make
up 26% and 12% of the total [CII] emission of the galaxy VC8326, respectively.
Together, therefore, they contain about 38% of the total [CII] flux. The two
components are clearly real components as they correspond to a fidelity of
100% (at the 4 $\sigma$ level). As seen in this section and discussed later in
the paper, this system is likely a merger - therefore the remaining emission
can be considered also in this case a diffuse gas component. However, since
that in this case the emission is picked up in all concatenations, it is
likely less diffuse than in DC8737.

Table 4: Resolved flux measurements of VC8326 Source | Flux (HR+MR) | Aperture Diameter
---|---|---
|  Jy km s-1 | kpc
Clump SE | $0.87\pm 0.15$ | $1.5\times 1.5$
Clump NW | $0.41\pm 0.13$ | $1.2\times 1.2$
Integrated galaxy | $3.3\pm 0.48$ | $4\times 4$

####  3.4.3 DC8187

DC8187 is constituted of 3 galaxies at about the same redshift - this with the
caveat of assuming observed emission is [CII]. The moment 0 map, shown in
Figure3, shows 2 of these galaxies (the Eastern and Central galaxy) are
currently interacting, with evidence of 2 clumps surrounding the central
galaxy. The Western source (as found in previous papers, see sect:2) has no
indication of ongoing merging activity, and is instead a potential future
merger.

The Western source is compact. The fluxes are 0.95, and 1.24 Jy km s-1, for
the MR+HR, and LR+MR+HR concatenations, respectively. Hence, the fluxes are
comparable within the error bounds and thus there is no evidence of flux loss
between resolutions - indicating that there is not a significant extended and
diffuse gas component.

The central source presents some flux loss between the HR+MR+LR and HR+MR
observations: the fluxes in these two different configurations are 5.3 and
4.37 Jy km s-1, respectively. This suggests that this object is constituted by
both a compact and an extended component.

The new observations clearly show that the central source appears to be
surrounded by two fainter clumps (as shown in Figure3). For this observation,
the S/N at which the fidelity reaches 60% is 3.6: the clump in the South East
is detected at 5$\sigma$, corresponding to a fidelity of 100%; the North
Western clump is detected at approximately 4 $\sigma$ \- indicating it is a
real object. In Table 5 the flux from the clumps is reported. The clumps make
up 12 and 5% of the total emission of the central galaxy, respectively - a
much smaller fraction than for the clumps in VC8326.

Table 5: Resolved flux measurements of DC8187 in HR+MR+LR concatenation Source | HR+MR+LR Flux | Aperture Diameter
---|---|---
|  Jy km s-1 | kpc
Galaxy C | $5.3\pm 0.71$ | $8.8\times 8.8$
Clump SW | $0.68\pm 0.23$ | $2.0\times 2.0$
Clump NE | $0.30\pm 0.13$ | $1.1\times 1.1$

The flux loss is less prominent in the Eastern source due to the poor
detection at the highest resolution and thus large errors associated with it.
However, like the central source there does appear to be flux loss between the
HR+MR and HR+MR+LR concatenations, (the fluxes are 3.05 and 3.54 Jy km s-1 for
the HR+MR and HR+MR+LR respectively).

The flux loss in both the Central and Eastern sources appears to indicate that
there is a diffuse gas component. This gas component is likely a consequence
of the merger, which is disrupting gas and leading to the formation of gas
bridges, extended emission and clumps seen in the moment 0 maps.

###  3.5 Continuum analysis and radial profiles

As previously mentioned, the 158$\mu$m continuum is detected in all three
systems. In this section, we compare the location and intensity of the FIR
emission and that of the [CII], also exploiting [CII] and continuum radial
profiles. The radial profiles are obtained integrating the signal ([CII] or
continuum) within circular apertures of increasing size (step 1 pixel),
starting with a circle that has the same radius as the major axis of the beam.
The profiles are centered on the barycenter of the [CII] emission (indicated
as a white cross in Figures 1, 2, and 3): note that while for DC8737 this
corresponds as well with the peak of the [CII] and continuum emission, for
VC8326 this barycenter does not correspond to the peak of the [CII] emission
(that is concentrated on the two clumps).

As [CII] is often used as a tracer for molecular gas (Zanella et al., 2018),
we convert the [CII] fluxes, for each aperture, to the molecular gas mass
surface density, using equation 2:

$\Sigma_{\text{gas
}}=\alpha_{[\mathrm{CII}]}\frac{1}{D_{A}^{2}}\left(1.04\times
10^{-3}\frac{\mathrm{L}_{\odot}\mathrm{s}}{\mathrm{GHz}\mathrm{Mpc}^{2}\mathrm{Jy}\mathrm{km}}\right)D_{L}^{2}v_{\text{obs
}}m_{[\mathrm{CII}]},$ (2)

where $\alpha_{[\mathrm{CII}]}$ is the conversion factor between [CII] and
molecular gas mass (31 $M_{\odot}$/$L_{\odot}$, Zanella et al. (2018)),
$D_{L}$ is the luminosity distance, $\nu_{[\mathrm{obs}]}$is the observed
frequency, $\ m_{[\mathrm{CII}]}$ is the flux in Jy of [CII], and $D_{A}$ is
the angular size distance.

The continuum at 158 $\mu$m is assumed to represent obscured star formation:
we convert the continuum flux to surface density of obscured star formation
using equation 3:

$\Sigma_{\mathrm{SFR}_{\mathrm{IR}}}=\kappa_{\mathrm{IR}}\frac{1}{D_{A}^{2}}\frac{L_{\mathrm{IR}}}{L_{158}}v_{[CII]}\frac{4\pi
D_{L}^{2}}{1+z}m_{158},$ (3)

where $L_{[\mathrm{IR}]}$/$L_{[\mathrm{158}]}$ is the ratio between the total
infrared luminosity and the 158 $\mu$m rest-frame monochromatic luminosity (we
use the value of 1/0.113 as found by Béthermin et al. (2020)),
$\nu_{[\mathrm{CII}]}$ is the rest-frame velocity of [CII],
$m_{[\mathrm{158}]}$ is the flux in Jy of the continuum and
$\kappa_{[\mathrm{IR}]}$ is the conversion factor $1.47\times 10^{-10}$
$M_{\odot}$/$L_{\odot}$.

The surface density of molecular gas mass and obscured star formation (taken
for each aperture) are plotted against the physical scale of the corresponding
aperture. This gives a radial profile of emission of gas and obscured star
formation for each Galaxy. These are shown in Figure4 and are discussed in the
next subsections.

Figure 4: Radial profiles of the galaxies in this paper with both [CII] and
Continuum emission (using the HR+MR+LR concatenation). The [CII] emission was
converted to a molecular gas surface density and the continuum to an obscured
SFR. The aperture of which the galaxies fluxes in sectopm3.4were taken in the
HR+MR+LR concatenation is shown by the blue line

####  3.5.1 DC8737

As discussed in the previous section, DC8737 consists of a compact and an
extended [CII] component. From the third panel in the top row of Figure 1, it
is clear that the continuum emission is aligned with the [CII] emission, and
is much more compact than the [CII] one. In addition, from the top left panel
of Figure 4, we again see that the obscured SFR, traced by continuum, is more
compact than the molecular gas, traced by [CII].

####  3.5.2 VC8326

VC8326 consists of two distinct components in [CII], which could be two
merging galaxies. However, the continuum shows a radically different
morphology: as shown by the right panel of the top row in Figure2, the
continuum emission is divided into 3 components, one of which is located at
the center of the galaxy, near the barycenter of the [CII] emission. The other
two continuum components (one in the south-western clump and one further south
of the galaxy) are both detected at 5 sigma - corresponding to a fidelity of
100%. It is worth noting, as it can be seen in the central panel of the top
row in Figure 2, that this system is also well detected in the F814W image by
HST and in the K-band image by UVISTA, tracing the UV and optical rest-frame,
respectively. Both emissions significantly differ from the appearance of the
[CII] or continuum emission, as they show only a single component well
centered on the barycenter of the [CII] emission, with no evidence for
substructure.

From Figure 4, it is again evident that the [CII] and continuum emission are
offset with respect to each other: while the continuum emission monotonically
decreases from the center, that has been chosen as the barycenter of the [CII]
emission (white cross in the top right panel of Figure 2), the [CII] emission
increases for increasingly large apertures until the flux from the two main
peaks is included (for the aperture with a 2.5 kpc radius), and then starts
decreasing.

This presence of continuum emission in the center of the system suggests that
some amount of obscured star-formation is taking place, although no [CII]
emission is detected there. The south-western [CII] component also shows
continuum emission, suggestive of a large amount of obscured star formation,
that could be feeding from the large gas reservoir traced by [CII].

Finally, there is evidence of a third continuum component with no
corresponding [CII] emission (nor HST or VISTA), indicating that there may be
an additional component with recent star-formation (which has used all its
molecular gas) and with a large quantity of dust (explaining the lack of
visible HST emission).

####  3.5.3 DC8187

As shown by the third panel in the top row of Figure 3, continuum emission is
only significantly detected for the central and western galaxies, while there
is no detection in the Eastern source.

The continuum emission in the central component is coming from 3 clumps and a
central region, as shown in the top row of 3, and it is concentrated within an
area of $\sim$ 3 kpc in diameter, as shown by the radial profile in Figure 4:
at larger radii, the profile drops off - before becoming noise dominated. The
clump to the north east is detected at 3 $\sigma$ which is below a 60%
fidelity. However the clumps to the south and north-west are detected at 4
$\sigma$ corresponding to 80% fidelity - indicating they are likely real. UV
and optical rest-frame emission, traced by HST/ACS F814W and K band UVISTA,
are very well detected for the central component, and aligned with the [CII]
emission.

The western galaxy is quite interesting: it is extremely compact - less than
$\sim$2 kpc in diameter (as it can be seen in both in Figure 3 and 4), both in
[CII] and continuum, that have similar profiles.

###  3.6 Position-Velocity Diagrams

Position-Velocity (PV) diagrams can give important insights on the kinematic
properties of galaxies, and therefore were produced for each individual source
source. PV diagrams are primarily extracted on the HR+MR concatenations or
(except for DC8737 where the HR+MR+LR concatenation is used due to the
extended nature of the source). The reason for this choice is that this
configuration offers the best compromise between resolution and signal-to-
noise.

PV slits were selected to go through the peak emission for each galaxy - with
the slit oriented to, preferentially, go through extended emission or clumps
(where detected); a slit width of the average of the minor and major axis of
the beam was chosen for all sources. All PV diagrams have contours at
fractions of 0.7, 0.8 and 0.9 times of the peak flux in the PV diagram itself.
The contours were drawn with this method instead of using the RMS as the RMS
is variable within each channel. PV diagrams are shown in the bottom right
panels of Figures 1, 2 and 3 and will be analysed in the next sections to
constrain the kinematics of these galaxies.

###  3.7 Spectra

Spectra around the channels containing the [CII] emission were produced for
each source (including the individual components of DC8187). Spectra were
generated using aperture photometry on each individual channel in the data-
cube. The spectra presented in Figure 4 are extracted in the HR+MR
concatenation, with the exception of DC8737 which was obtained from the
HR+MR+LR configuration, to cover both the extended and compact components. The
HR+MR concatenations of all sources have a channel width of 6.75km/s, the
HR+MR+LR of DC8737 has a channel width of 13.5km/s. For all sources the
apertures are the same as in Section 3.4.

For each source, Gaussian models were fit to the data, including allowing the
possibility of multiple Gaussians to fit the data. For all objects but DC8737,
one single Gaussian was enough to reproduce the emission. For DC8737, some
excess emission around +400 km/s was detected, and therefore we added a second
Gaussian around that velocity to improve the fit. The middle panel of Figure 4
shows that the excess emission at $\sim$+400 km/s is significant at $\sim$3
$\sigma$

Figure 5: Spectra of [CII]. The top panel shows the the spectrum for DC8737
along with the residuals (middle panel) when the Gaussian fit is subtracted
from the data. The bottom 3 panels show from left to right: The spectra for
VC8327, DC8187 (Central and East galaxies) and finally DC8737 (West galaxy).

##  4 Outflows and Velocity Groups - DC8737

Outflows of molecular gas are identified as additional broad emissions,
usually offset with respect to the main emission (i.e. Ginolfi et al. (2020a),
Herrera-Camus et al. (2021b)). The presence of outflows would imply that the
total line emission would be best fit by 2 or 3 Gaussians as opposed to a
single Gaussian component. As seen in the top panel of Figure 5, for DC8737
there is evidence of a broad component at $\sim$400 km/s, in addition to the
global emission from the source. In the middle panel of Figure 5 we show the
residual emission when the main Gaussian is subtracted from the total
spectrum: the additional emission at $\sim$+400 km/s is evident as a clear
peak that has an integrated S/N of 10.8 (with a peak value of 3 $\sigma$). We
note that there is a second, much smaller peak at $\sim-$400 km/s, with an
integrated S/N of 3.6 only. This peak therefore cannot be conclusively
distinguished as genuine emission from noise. The fact the two peaks at +/-
400km/s are of different S/N, and only one is fit by a broad component, is not
consistent with the typical scenario of biconical outflow; it would appear
that we detect with high significance only the side of the outflow receding
from us and not the approaching side. Although it must be noted orientation
effects are such that even in the case of biconical outflows - only one wing
may be observed. The source of this outflow could be the intense star
formation identified in the center of DC8737, we cannot rule out the presence
of an Active Galactic Nuclei (AGN) which could also produce such outflows.

We present as well a second scenario which is consistent with the observed
second component at $\sim$+400 km/s: a merging object falling onto the central
one at that velocity would be observed an additional emission component,
offset with respect to the main one. In Figure6 a moment 0 map of only the
channels that contribute to the broad component is shown. The map displays a
0.3” offset to the South-West compared to the total [CII] emission -
comparable to the ALMA beam - further supporting the merger scenario compared
to the outflow scenario. However, it is interesting to note that this alleged
merging component is not detected neither in the HST/ACS F814W image nor in
the K-band UVISTA one (the only emission in those bands is close to the
barycenter of the [CII] emission, see Figure 1). For the UVISTA this is
expected due to the large PSF (0.7 $\arcsec$).

By looking at the [CII] moment 0 map for the HR+MR configuration, in the top
left panel of Figure 1, there is evidence of two smaller additional
components, one to the West and the other to the North of the central region.
These are well detected at over 4 $\sigma$\- corresponding to a fidelity of
100% - indicating that they are real components. The moment 1 maps, reported
in the bottom left panel of the same Figure, indicate that these regions both
move at an uniform velocity of -200km/s with respect to the main component -
indicating that they could be other minor mergers or simply clumps of star-
formation within the main object. Upcoming JWST data from the COSMOS-Web
survey Casey et al. (2023) and with NIRSpec IFU (P.I. A. Faisst) will allow
this to be conclusively determined. This additional components support the
hypothesis that DC8737 consists of multiple merging components - explaining
its high velocity dispersion - see a discussion on a similar object Ginolfi et
al. (2020b).

Figure 6: Map of [CII] for the broad component of DC8737. The contours from
$\pm 3,4,5,\cdots$ are shown in white. The [CII] map created over all line
emission is shown by red

##  5 Discussion

In this section we wrap up all the information that we gathered for the
objects in the sample and we discuss the properties of each one individually.

###  5.1 DC8737

Low resolution observations (from the original ALPINE survey) of DC8737, the
brightest galaxy in the ALPINE survey in [CII], showed no signs of rotation,
and suggested that this object is an extended dispersion dominated galaxy.
However, the addition of high resolution observations that we present in this
paper show that this object is definitely more complex than initially thought.

In Section 3.4 we showed that approximately 50% of emission is traced by the
HR+MR observations compared to HR+MR+LR. This indicates that there appears to
be 2 components in [CII]: a compact and bright central region, and a fainter,
more extended component. The higher resolution concatenations only detect the
central, compact region - indicating higher resolutions trace the compact
emission as expected; however, these higher resolution concatenations lack the
integration time needed to detect the extended components above noise.

The extended [CII] component is interpreted as a diffuse gas component,
similarly to Fujimoto et al. (2020). The compact central region is also
detected in HST F160w imaging and in the FIR continuum - indicating a highly
star forming center, as presented in Béthermin et al. (2023, submitted).

The PV diagram and the HR+MR+LR moment 1 map, shown in Fig. 1, show that the
galaxy appears dispersed (with multiple velocities), with no evidence of
ordered velocity gradients that could be suggesting rotation. These groups are
concentrated in the central, compact region detected in the high resolution
observations. This indicates that the central region could be made up of
distinct velocity components. In, the HR+MR map there is evidence for 2
components to the North and West of the central region at a velocity of
($<-200\,{\rm km/s}$) this could be indicative of galaxy components (i.e.
minor merger) as discussed in Section 4.

DC8737 appears to show evidence of an additional [CII] component at $\sim$+400
km/s, that could be interpreted either as outflowing gas or as a minor merger,
as discussed in detail in Section 4. In support of the outflow scenario, we
highlight that the FIR continuum, [CII] and HST maps show intense star
formation occurring in the dusty central region, that is expected to drive
outflows (Ginolfi et al., 2020a), although we note the presence of an AGN
cannot be excluded for this galaxy. Moreover, the evidence for an extended
[CII] halo around the more compact central emission, is in agreement with the
recent semi-analytical model by Pizzati et al. (2022), in which star-formation
driven outflows can produce these extended gas halos. In a recent study,
Romano et al. (2023) traced outflowing gas in a sample of local dwarf galaxies
(analogs of high-z SFGs) from [CII] emission: they found that [CII] is $\sim$2
times more extended than UV emission, with 40% of the gas possibly pushed
outside of the galaxies by the otuflow (thus enriching the CGM around those
galaxies).

If the additional component were instead due to a minor merger, together with
the evidence of fainter, but statistically significant, [CII] components in
the moment 0 map (Figure 1), this could be an indication of a very turbulent
environment in which smaller components (or minor mergers) are continuously
falling onto the main central galaxy, determining the very large velocity
dispersion of the system.

###  5.2 VC8326

VC8326 is the most complex source in the sample. There is consistency in [CII]
flux between the 3 concatenations for VC8327 - although 62% of the flux is
located outside of compact regions (see Section 3.4). The fact the [CII] flux
from this underlying component is detected in all concatenations suggests that
it is less diffuse than the [CII] halo in DC8737. Both concatenations show 2
clear compact clumps in [CII] - which are clearly real emission. For VC8326,
the low resolution observations from ALPINE was unable to resolve the two
clumps and the [CII] emission appeared to come from a single component, which
was dispersion dominated; the high resolution data is therefore crucial to
reveal the real nature of this system.

Understanding whether VC8326 is composed of 2 merging galaxies or rather a
single galaxy with 2 in-situ clumps is not trivial. The separation of the two
[CII] components, around 5 kpc, is much larger than typical galaxies at the
same redshift (Ito et al., 2023), this suggests that they could indeed be two
individual merging galaxies. The South-Western clump is dusty (seen by
emission in the continuum maps in 2). The size of the clump is consistent with
smaller galaxies (i.e. DC8187 W) which could indicate this is an individual
galaxy. There is [CII] located between the two main [CII]components which is
comparable to the diffuse [CII] in the rest of the galaxy - the moment 0 map
shows an unresolved dip in [CII] which is not real and is an effect from the
cleaning process. There is strong HST emission in the center (and dust
continuum emission), this can be explained by merger induced star formation.

On the other hand there is some evidence against this scenario. As shown by
the middle panel of Figure 2 rest-frame UV observations from HST indicate
intense star formation in the central region, exactly mid-way between the two
[CII] compact regions); moreover, this is corroborated by a tentative
detection of continuum in the center (top right panel of Figure 2); also,
there is no HST emission from the clumps and K-band emission, tracing the
rest-frame optical. Therefore it is very possible that VC8326 is indeed a
large galaxy in formation, with two main clumps of star-formation. The
brightest is very star-forming and therefore detected both in [CII] and also
in the continuum, while the faintest is not detected in continuum: this could
be simply because this latter is less star-forming, and therefore under the
ALMA detection limit for the continuum, or it could just be younger and
therefore less dust rich.

In any case, we highlight again the presence of a third clump South-West of
the [CII] emission that is only detected in continuum (and not in [CII]): this
could after-all be an old clump that exhausted all the gas within it, or a
less massive dust-rich merging component.

###  5.3 DC8187

The ALPINE data of this triple system evidenced that it was made of three
galaxies, but the poor resolution prevented us from analysing their internal
properties. The addition of the high resolution data in this paper allows us
to begin resolving the three sources, leading, for the first time, for clumps
to be detected in this system.

In particular, the central galaxy (that is currently interacting with the
Eastern source) exhibits [CII] clumps surrounding the main component that were
not identified with the original ALPINE data (top left panel of Figure 3).
These clumps exhibit a clear separation in velocity from the rest of the
galaxy, as demonstrated by the moment 1 map Figure(3). These clumps are denser
and thus more highly star forming compared to the underlying disk. They are
also less bright and massive than the big clumps seen in VC8326, as they make
up only a tiny fraction of the total [CII] emission. This is consistent with
simulations (Pallottini et al., 2022b), which suggest that mergers may indeed
induce clumpy morphologies.

There is also evidence of 2 clumps detected in the Continuum, as shown in the
top right panel of Figure 3, and discussed in Sec.3.5. These have high
fidelity and therefore appear to be real components. It is important to
highlight that they are not coincident with the [CII] clumps - this could
indicate that clumps with dust obscured star formation are also produced in
merging galaxies

Both the Central and Eastern sources appear to have compact and faint
components. This is much more evident in DC8187 (C) which appears to have a
dusty star-forming, compact center (detected in the continuum and HST, see
Figure 3) along with a faint diffuse gas component.

The Western most galaxy (which is not strongly interacting with the other two)
is compact in [CII] (with an approximate size of $0.3\arcsec$ size - 1.1kpc)
when observed in the HR+MR configuration. The velocity map (bottom left panel
of Figure 3), shows a velocity gradient consistent with rotation. There is no
signs of rotation from the PV diagram - likely due to insufficient signal to
noise leading to a poor detection in the PV diagram. If this is truly a
rotating disk, this presents a case of a low mass disk z$\sim$4.5.

##  6 Conclusions and summary

Using new 0.15” resolution ALMA observations in band 7 we have gained new
insights into the nature of 3 primordial main-sequence galaxy systems (once of
which is composed of 3 galaxies) at z$\sim$4.5, that were previously studied
as part of the ALPINE survey Le Fèvre et al. (2020); Béthermin et al. (2020);
Faisst et al. (2020). The new high resolution data, although not reaching the
sensitivity that was originally requested, reveal many more details than the
original ALPINE data. The main conclusions are as follows:

  1. 1.

VC8326 and DC8737 are the first and third brightest [CII] systems in ALPINE,
sharing similar morpho-kinematic properties in the low resolution data.
However, when observed at high resolution, they turn out to be very different.
DC8737 shows a very bright and concentrated [CII] emission in the center,
surrounded by a [CII] halo and a few [CII] fainter clumps. A [CII] component
offset by 400 km/s is also revealed, but we could not conclude whether this
indicates an outflow or a merging component. The continuum emission is also
aligned with [CII] (although much less extended).

  2. 2.

VC8326, on the other hand, is resolved by the high-resolution data presented
in this paper into two [CII] regions, embedded into an underlying broader
component (not as diffuse as the one around DC8737). It is unclear whether
these two main [CII] components can be considered two merging galaxies (as
suggested by their size and separation) or they are large star-forming regions
within the same galaxy (as suggested by the morphology of the UV and optical
rest-frame light, revealed by HST and UVISTA, respectively). Moreover, for
this object, the continuum emission is also spatially offset with respect to
the [CII] emission (except for one of the clumps).

  3. 3.

DC8187, that is composed of 3 galaxies, two of which are in the process of
interacting with each other, gave us the opportunity to study the effect of
mergers on the internal properties of the interacting galaxies. We revealed
[CII] and dust bright clumps that were not identified by the low resolution
ALPINE data, and that could be induced by the merger activity, as suggested by
recent simulations.

  4. 4.

Remarkably, we also revealed that the third non-interacting galaxy in DC8187
is a compact object, very likely rotating. If confirmed by upcoming JWST
NIRSpec-IFU observations, this could be one of the most distant settled disk
ever identified.

The addition of these high resolution observations has interesting
implications for our knowledge of galaxy evolution in the first billion year
after the Big Bang. The most striking point is that there is evidence of
mergers or components in both dispersion dominated galaxies in this sample.
These observations indicate that mergers can be more frequent than what found
in the ALPINE survey (Romano et al., 2021b). Whilst our sample remains small,
a higher merger fraction would indicate that mergers remain important to the
mass assembly of galaxies in this early epoch. It must be noted that the
sources identified as potential major mergers (VC8326 and DC8187) are located
in the center and edge, respectively, of the J1001+0220 protocluster (Lemaux
et al. (2018), Staab et al. (2023, submitted) - highlighting how environment
may be important in shaping galaxy evolution at such redshifts.

We also identify a potential low mass disk galaxy at redshift z$\sim$4.5,
DC8187(W), in agreement with recent JWST observations: Ferreira et al. (2022)
have indicated that disk galaxies might settle much earlier than previously
thought, with a larger than expected number of disks in the very early
Universe. Furthermore since DC8187(W) is the galaxy with the lowest gas mass
in [CII], among those int the triple merger system: this indicates that disks
are not necessarily the most mature or massive galaxies at these redshifts.

We also identify clumps in [CII] for a major merging system DC8187. This
supports simulations which show that major mergers can induce clump formation
(Pallottini et al., 2022b). In our observations we observe two distinct
categories of clumps: those only visible in [CII] and those only visible in
the continuum, but neither of these clumps are visible in HST observations of
this galaxy system.

We interpret these clumps are sites of merger-induced star formation: the two
categories of clumps are explained by the fact that as time passes, molecular
gas is being consumed in the production of stars (and hence dust). Therefore,
the clumps observed in [CII] might be younger clumps with higher quantities of
molecular gas, but still relatively low star-formation activity, whereas the
continuum clumps are older clumps where the molecular gas has been fully
consumed, but which have larger dust content. The presence of two types of
clump indicates star formation may occur over a short period - otherwise we
would only observe one type of clump. Simulations Bournaud et al. (2014);
Mandelker et al. (2014), of clumps indicates that clumps with short bursts of
star formation can survive stellar feedback for longer periods (¿500Myr). This
indicates that these clumps could survive timescales long enough for them to
migrate to the central component – and be involved the evolution of the galaxy
into a settled disk.

Planned JWST observations with NIRSpec/IFU will allow us to perform spatially
resolved SED fitting to understand the properties of these clumps and
determine if they are comparable to clumps observed at cosmic noon.

## Acknowledgements

This paper makes use of the following ALMA data: ADS/JAO.ALMA#2019.1.00226.S
ALMA is a partnership of ESO (representing its member states), NSF (USA) and
NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ.

E.I. acknowledge funding by ANID FONDECYT Regular 1221846

M.B. gratefully acknowledges support from the ANID BASAL project FB210003 and
from the FONDECYT regular grant 1211000.

G.E.M. acknowledges the Villum Fonden research grant 13160 “Gas to stars,
stars to dust: tracing star formation across cosmic time,” grant 37440, “The
Hidden Cosmos,” and the Cosmic Dawn Center of Excellence funded by the Danish
National Research Foundation under the grant No. 140

C.H acknowledges The Flatiron Institute and is supported by the Simons
Foundation.

M.R. acknowledges support from the Narodowe Centrum Nauki
(UMO-2020/38/E/ST9/00077) and support from the Foundation for Polish Science
(FNP) under the program START 063.2023.

## Data Availability

All raw data available from the ALMA archive.

## References

  * Béthermin et al. (2020) Béthermin M., Fudamoto Y., Ginolfi M., et al. 2020, http://dx.doi.org/10.1051/0004-6361/202037649 A&A, https://ui.adsabs.harvard.edu/abs/2020A&A…643A…2B 643, A2
  * Bolatto et al. (2013) Bolatto A. D., Wolfire M., Leroy A. K., 2013, http://dx.doi.org/10.1146/annurev-astro-082812-140944 ARA&A, https://ui.adsabs.harvard.edu/abs/2013ARA&A..51..207B 51, 207
  * Bournaud et al. (2014) Bournaud F., Perret V., Renaud F., et al. 2014, http://dx.doi.org/10.1088/0004-637X/780/1/57 ApJ, https://ui.adsabs.harvard.edu/abs/2014ApJ…780…57B 780, 57
  * Carilli & Walter (2013) Carilli C. L., Walter F., 2013, http://dx.doi.org/10.1146/annurev-astro-082812-140953 ARA&A, https://ui.adsabs.harvard.edu/abs/2013ARA&A..51..105C 51, 105
  * Casey et al. (2023) Casey C. M., Kartaltepe J. S., Drakos N. E., et al. 2023, http://dx.doi.org/10.3847/1538-4357/acc2bc ApJ, https://ui.adsabs.harvard.edu/abs/2023ApJ…954…31C 954, 31
  * Cassata et al. (2020) Cassata P., Morselli L., Faisst A., et al. 2020, http://dx.doi.org/10.1051/0004-6361/202037517 A&A, https://ui.adsabs.harvard.edu/abs/2020A&A…643A…6C 643, A6
  * Claeyssens et al. (2023) Claeyssens A., Adamo A., Richard J., et al. 2023, http://dx.doi.org/10.1093/mnras/stac3791 MNRAS, https://ui.adsabs.harvard.edu/abs/2023MNRAS.520.2180C 520, 2180
  * D’Eugenio et al. (2023) D’Eugenio C., Daddi E., Liu D., Gobat R., 2023, http://dx.doi.org/10.1051/0004-6361/202347233 A&A, https://ui.adsabs.harvard.edu/abs/2023A&A…678L…9D 678, L9
  * De Looze et al. (2014) De Looze I., Cormier D., Lebouteiller V., et al. 2014, http://dx.doi.org/10.1051/0004-6361/201322489 A&A, https://ui.adsabs.harvard.edu/abs/2014A&A…568A..62D 568, A62
  * Dekel et al. (2009) Dekel A., Sari R., Ceverino D., 2009, http://dx.doi.org/10.1088/0004-637X/703/1/785 ApJ, https://ui.adsabs.harvard.edu/abs/2009ApJ…703..785D 703, 785
  * Dessauges-Zavadsky et al. (2020) Dessauges-Zavadsky M., Ginolfi M., Pozzi F., et al. 2020, http://dx.doi.org/10.1051/0004-6361/202038231 A&A, https://ui.adsabs.harvard.edu/abs/2020A&A…643A…5D 643, A5
  * Di Matteo et al. (2008) Di Matteo P., Bournaud F., Martig M., et al. 2008, http://dx.doi.org/10.1051/0004-6361:200809480 A&A, https://ui.adsabs.harvard.edu/abs/2008A&A…492…31D 492, 31
  * Di Teodoro & Fraternali (2015) Di Teodoro E. M., Fraternali F., 2015, http://dx.doi.org/10.1093/mnras/stv1213 MNRAS, https://ui.adsabs.harvard.edu/abs/2015MNRAS.451.3021D 451, 3021
  * Faisst et al. (2020) Faisst A. L., Schaerer D., Lemaux B. C., et al. 2020, http://dx.doi.org/10.3847/1538-4365/ab7ccd ApJS, https://ui.adsabs.harvard.edu/abs/2020ApJS..247…61F 247, 61
  * Ferreira et al. (2022) Ferreira L., Adams N., Conselice C. J., et al. 2022, http://dx.doi.org/10.3847/2041-8213/ac947c ApJ, https://ui.adsabs.harvard.edu/abs/2022ApJ…938L…2F 938, L2
  * Finkelstein et al. (2022) Finkelstein S. L., Bagley M. B., Arrabal Haro P., et al. 2022, http://dx.doi.org/10.3847/2041-8213/ac966e ApJ, https://ui.adsabs.harvard.edu/abs/2022ApJ…940L..55F 940, L55
  * Förster Schreiber & Wuyts (2020) Förster Schreiber N. M., Wuyts S., 2020, http://dx.doi.org/10.1146/annurev-astro-032620-021910 ARA&A, https://ui.adsabs.harvard.edu/abs/2020ARA&A..58..661F 58, 661
  * Förster Schreiber et al. (2009) Förster Schreiber N. M., Genzel R., Bouché N., et al. 2009, http://dx.doi.org/10.1088/0004-637X/706/2/1364 ApJ, https://ui.adsabs.harvard.edu/abs/2009ApJ…706.1364F 706, 1364
  * Fujimoto et al. (2020) Fujimoto S., Silverman J. D., Bethermin M., et al. 2020, http://dx.doi.org/10.3847/1538-4357/ab94b3 ApJ, https://ui.adsabs.harvard.edu/abs/2020ApJ…900….1F 900, 1
  * Ginolfi et al. (2020a) Ginolfi M., Jones G. C., Béthermin M., et al. 2020a, http://dx.doi.org/10.1051/0004-6361/201936872 A&A, https://ui.adsabs.harvard.edu/abs/2020A&A…633A..90G 633, A90
  * Ginolfi et al. (2020b) Ginolfi M., Jones G. C., Béthermin M., et al. 2020b, http://dx.doi.org/10.1051/0004-6361/202038284 A&A, https://ui.adsabs.harvard.edu/abs/2020A&A…643A…7G 643, A7
  * Harikane et al. (2023) Harikane Y., Ouchi M., Oguri M., et al. 2023, http://dx.doi.org/10.3847/1538-4365/acaaa9 ApJS, https://ui.adsabs.harvard.edu/abs/2023ApJS..265….5H 265, 5
  * Herrera-Camus et al. (2021a) Herrera-Camus R., Förster Schreiber N., Genzel R., et al. 2021a, http://dx.doi.org/10.1051/0004-6361/202039704 A&A, https://ui.adsabs.harvard.edu/abs/2021A&A…649A..31H 649, A31
  * Herrera-Camus et al. (2021b) Herrera-Camus R., Förster Schreiber N., Genzel R., et al. 2021b, http://dx.doi.org/10.1051/0004-6361/202039704 A&A, https://ui.adsabs.harvard.edu/abs/2021A&A…649A..31H 649, A31
  * Ito et al. (2023) Ito K., Valentino F., Brammer G., et al. 2023, http://dx.doi.org/10.48550/arXiv.2307.06994 arXiv e-prints, https://ui.adsabs.harvard.edu/abs/2023arXiv230706994I p. arXiv:2307.06994
  * Jones et al. (2020) Jones G. C., Béthermin M., Fudamoto Y., et al. 2020, http://dx.doi.org/10.1093/mnrasl/slz154 MNRAS, https://ui.adsabs.harvard.edu/abs/2020MNRAS.491L..18J 491, L18
  * Jones et al. (2021) Jones G. C., Vergani D., Romano M., et al. 2021, http://dx.doi.org/10.1093/mnras/stab2226 MNRAS, https://ui.adsabs.harvard.edu/abs/2021MNRAS.507.3540J 507, 3540
  * Kennicutt (1998) Kennicutt Robert C. J., 1998, http://dx.doi.org/10.1086/305588 ApJ, https://ui.adsabs.harvard.edu/abs/1998ApJ…498..541K 498, 541
  * Koekemoer et al. (2007) Koekemoer A. M., Aussel H., Calzetti D., et al. 2007, http://dx.doi.org/10.1086/520086 ApJS, https://ui.adsabs.harvard.edu/abs/2007ApJS..172..196K 172, 196
  * Kohandel et al. (2019) Kohandel M., Pallottini A., Ferrara A., et al. 2019, http://dx.doi.org/10.1093/mnras/stz1486 MNRAS, https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.3007K 487, 3007
  * Le Fèvre et al. (2020) Le Fèvre O., Béthermin M., Faisst A., et al. 2020, http://dx.doi.org/10.1051/0004-6361/201936965 A&A, https://ui.adsabs.harvard.edu/abs/2020A&A…643A…1L 643, A1
  * Lemaux et al. (2018) Lemaux B. C., Le Fèvre O., Cucciati O., et al. 2018, http://dx.doi.org/10.1051/0004-6361/201730870 A&A, https://ui.adsabs.harvard.edu/abs/2018A&A…615A..77L 615, A77
  * Litke et al. (2019) Litke K. C., Marrone D. P., Spilker J. S., et al. 2019, http://dx.doi.org/10.3847/1538-4357/aaf057 ApJ, https://ui.adsabs.harvard.edu/abs/2019ApJ…870…80L 870, 80
  * Madau & Dickinson (2014) Madau P., Dickinson M., 2014, http://dx.doi.org/10.1146/annurev-astro-081811-125615 ARA&A, https://ui.adsabs.harvard.edu/abs/2014ARA&A..52..415M 52, 415
  * Maiolino et al. (2023) Maiolino R., Uebler H., Perna M., et al. 2023, http://dx.doi.org/10.48550/arXiv.2306.00953 arXiv e-prints, https://ui.adsabs.harvard.edu/abs/2023arXiv230600953M p. arXiv:2306.00953
  * Mandelker et al. (2014) Mandelker N., Dekel A., Ceverino D., et al. 2014, http://dx.doi.org/10.1093/mnras/stu1340 MNRAS, https://ui.adsabs.harvard.edu/abs/2014MNRAS.443.3675M 443, 3675
  * Naidu et al. (2022) Naidu R. P., Oesch P. A., van Dokkum P., et al. 2022, http://dx.doi.org/10.3847/2041-8213/ac9b22 ApJ, https://ui.adsabs.harvard.edu/abs/2022ApJ…940L..14N 940, L14
  * Oteo et al. (2018) Oteo I., Ivison R. J., Dunne L., et al. 2018, http://dx.doi.org/10.3847/1538-4357/aaa1f1 ApJ, https://ui.adsabs.harvard.edu/abs/2018ApJ…856…72O 856, 72
  * Pallottini et al. (2022a) Pallottini A., Ferrara A., Gallerani S., et al. 2022a, http://dx.doi.org/10.1093/mnras/stac1281 MNRAS, https://ui.adsabs.harvard.edu/abs/2022MNRAS.513.5621P 513, 5621
  * Pallottini et al. (2022b) Pallottini A., et al., 2022b, http://dx.doi.org/10.1093/mnras/stac1281 Mon. Not. Roy. Astron. Soc., 513, 5621
  * Pizzati et al. (2022) Pizzati E., Ferrara A., Pallottini A., et al. 2022, http://dx.doi.org/10.1093/mnras/stac3816 Monthly Notices of the Royal Astronomical Society, 519, 4608
  * Popping et al. (2023) Popping G., Shivaei I., Sanders R. L., et al. 2023, http://dx.doi.org/10.1051/0004-6361/202243817 A&A, https://ui.adsabs.harvard.edu/abs/2023A&A…670A.138P 670, A138
  * Roman-Oliveira et al. (2023) Roman-Oliveira F., Fraternali F., Rizzo F., 2023, http://dx.doi.org/10.1093/mnras/stad530 MNRAS, https://ui.adsabs.harvard.edu/abs/2023MNRAS.521.1045R 521, 1045
  * Romano et al. (2021a) Romano M., Cassata P., Morselli L., et al. 2021a, http://dx.doi.org/10.1051/0004-6361/202141306 A&A, https://ui.adsabs.harvard.edu/abs/2021A&A…653A.111R 653, A111
  * Romano et al. (2021b) Romano M., Cassata P., Morselli L., et al. 2021b, http://dx.doi.org/10.1051/0004-6361/202141306 A&A, https://ui.adsabs.harvard.edu/abs/2021A&A…653A.111R 653, A111
  * Romano et al. (2023) Romano M., Nanni A., Donevski D., et al. 2023, http://dx.doi.org/10.1051/0004-6361/202346143 A&A, https://ui.adsabs.harvard.edu/abs/2023A&A…677A..44R 677, A44
  * Schreiber et al. (2015) Schreiber C., Pannella M., Elbaz D., et al. 2015, http://dx.doi.org/10.1051/0004-6361/201425017 A&A, https://ui.adsabs.harvard.edu/abs/2015A&A…575A..74S 575, A74
  * Simons et al. (2019) Simons R. C., Kassin S. A., Snyder G. F., et al. 2019, http://dx.doi.org/10.3847/1538-4357/ab07c9 ApJ, https://ui.adsabs.harvard.edu/abs/2019ApJ…874…59S 874, 59
  * Song et al. (2016) Song M., Finkelstein S. L., Ashby M. L. N., et al. 2016, http://dx.doi.org/10.3847/0004-637X/825/1/5 ApJ, https://ui.adsabs.harvard.edu/abs/2016ApJ…825….5S 825, 5
  * Tasca et al. (2015) Tasca L. A. M., Le Fèvre O., Hathi N. P., et al. 2015, http://dx.doi.org/10.1051/0004-6361/201425379 A&A, https://ui.adsabs.harvard.edu/abs/2015A&A…581A..54T 581, A54
  * Tomczak et al. (2016) Tomczak A. R., Quadri R. F., Tran K.-V. H., et al. 2016, http://dx.doi.org/10.3847/0004-637X/817/2/118 ApJ, https://ui.adsabs.harvard.edu/abs/2016ApJ…817..118T 817, 118
  * Troncoso et al. (2014) Troncoso P., Maiolino R., Sommariva V., et al. 2014, http://dx.doi.org/10.1051/0004-6361/201322099 A&A, https://ui.adsabs.harvard.edu/abs/2014A&A…563A..58T 563, A58
  * Vallini et al. (2013) Vallini L., Gallerani S., Ferrara A., Baek S., 2013, http://dx.doi.org/10.1093/mnras/stt828 MNRAS, https://ui.adsabs.harvard.edu/abs/2013MNRAS.433.1567V 433, 1567
  * Vizgan et al. (2022) Vizgan D., Greve T. R., Olsen K. P., et al. 2022, http://dx.doi.org/10.3847/1538-4357/ac5cba ApJ, https://ui.adsabs.harvard.edu/abs/2022ApJ…929…92V 929, 92
  * Walter et al. (2011) Walter F., Weiß A., Downes D., Decarli R., Henkel C., 2011, http://dx.doi.org/10.1088/0004-637X/730/1/18 ApJ, https://ui.adsabs.harvard.edu/abs/2011ApJ…730…18W 730, 18
  * Walter et al. (2016) Walter F., Decarli R., Aravena M., et al. 2016, http://dx.doi.org/10.3847/1538-4357/833/1/67 ApJ, https://ui.adsabs.harvard.edu/abs/2016ApJ…833…67W 833, 67
  * Zanella et al. (2018) Zanella A., Daddi E., Magdis G., et al. 2018, http://dx.doi.org/10.1093/mnras/sty2394 MNRAS, https://ui.adsabs.harvard.edu/abs/2018MNRAS.481.1976Z 481, 1976
  * Zanella et al. (2021) Zanella A., Pallottini A., Ferrara A., et al. 2021, http://dx.doi.org/10.1093/mnras/staa2776 MNRAS, https://ui.adsabs.harvard.edu/abs/2021MNRAS.500..118Z 500, 118
  * Zhang et al. (2019) Zhang H., Primack J. R., Faber S. M., et al. 2019, http://dx.doi.org/10.1093/mnras/stz339 MNRAS, https://ui.adsabs.harvard.edu/abs/2019MNRAS.484.5170Z 484, 5170
  * van der Wel et al. (2014) van der Wel A., Franx M., van Dokkum P. G., et al. 2014, http://dx.doi.org/10.1088/0004-637X/788/1/28 ApJ, https://ui.adsabs.harvard.edu/abs/2014ApJ…788…28V 788, 28


On the Quantization of Length in
Noncommutative Spaces

Muthukumar Balasundaram$\dagger$00footnotetext: corresponding
<EMAIL_ADDRESS>and Aamir<EMAIL_ADDRESS>

Department of Physics,
School of Physical, Chemical and Applied Sciences,
Pondicherry University, Puducherry-605014, India

###### Abstract

We consider canonical/Weyl-Moyal type noncommutative (NC) spaces with
rectilinear coordinates. Motivated by the analogy of the formalism of the
quantum mechanical harmonic oscillator problem in quantum phase-space with
that of the canonical-type NC 2-D space, and noting that the square of length
in the latter case is analogous to the Hamiltonian in the former case, we
arrive at the conclusion that the length and area are quantized in such an NC
space, if the area is expressed entirely in terms of length. We extend our
analysis to the 3-D case and formulate a ladder operator approach to the
quantization of length in 3-D space. However, our method does not lend itself
to the quantization of spacetime length in 1+1 and 2+1 Minkowski spacetimes if
the noncommutativity between time and space is considered. If time is taken to
commute with spatial coordinates and the noncommutativity is maintained only
among the spatial coordinates in 2+1 and 3+1 dimensional spacetime, then the
quantization of spatial length is possible in our approach.

##  1 Introduction

Noncommutative (NC) spacetime, which was first introduced by Snyder [1] in an
attempt to regulate the divergences in quantum field theories, has also been
introduced in various contexts [2, 3, 4, 5, 6, 7] and the literature in this
area is quite replete. To name a few, field theories [8, 9, 10], gravity
theories [11, 12, 13, 14, 15], and quantum mechanics [16, 17, 18, 19, 20, 21,
22, 23, 24] have all been considered with the background spacetime being
noncommutative. In another development, the spectral manifolds in NC geometry
is shown to exhibit the geometric analogue of Heisenberg commutation relation
involving the Dirac operator and the Feynman slash operator for real scalar
fields, leading to the quantization of volume [25]. In [15], it was shown from
the pure geometrical point of view that the NC Minkowski spacetime
parametrized with spherical or cylindrical coordinates has nontrivial NC
corrections to curvature tensors and curvature scalar.

Noncommutativity of spatial coordinates is related to the presence of a
minimal length in a system. This minimal length in turn is usually related to
the uncertainties in the distance measurements [3, 22]. Instead of relating
the minimal length with uncertainties, we propose in this work an approach in
which the actual square of the distance
$L^{2}=g_{ij}(y^{i}-z^{i})(y^{j}-z^{j})$ between any two points $y$ and $z$ in
a commutative flat-spacetime is promoted as an operator with the introduction
of the canonical/Weyl-Moyal type noncommutativity
$[\hat{y}^{i},\hat{y}^{j}]=i\theta^{\prime ij}=[\hat{z}^{i},\hat{z}^{j}]$
among the coordinate operators. Here $g_{ij}$ is taken to be a constant
diagonal metric of spacetime and $\theta^{\prime ij}$ is a constant and real
antisymmetric matrix. The operators $\hat{y}^{i}$ and $\hat{z}^{i}$ may be
taken either as the position operators of two particles or as the operators
corresponding to the positions at which fields are considered. The idea of
length as an operator has already been discussed in the literature in the
context of canonical quantum gravity [26]. We set up an algebra of operators
in such a way that the eigenvalues of the operator $\hat{L}^{2}$ can be raised
or lowered. The hint to such an approach is provided by an analogy of 2-D NC
space operator formalism with that of the quantum mechanical harmonic
oscillator problem. Taking $\hat{x}^{i}=\hat{y}^{i}-\hat{z}^{i}$ and assuming
that the coordinate operators of different particles commute, i.e.,
$[\hat{y}^{i},\hat{z}^{j}]=0$, we can define the operator corresponding to the
square of the distance as

$\displaystyle\hat{L}^{2}=g_{ij}\hat{x}^{i}\hat{x}^{j}$ (1)

with

$\displaystyle[\hat{x}^{i},\hat{x}^{j}]=i\theta^{ij},$ (2)

where $\theta^{ij}=2\theta^{\prime ij}$. If $g_{ij}=\mathrm{diag}(1,1)$, then
the operator $\hat{L}^{2}$ in such a 2-D NC space can be related to the
Hamiltonian of an appropriate harmonic oscillator. One important thing in such
an analogy is the set of ladder operators $\hat{a}_{-}$, the lowering
operator, and $\hat{a}_{+}=\hat{a}_{-}^{\dagger}$, the raising operator. If
$\hat{X}=\left(\begin{array}[]{l}\hat{x}^{1}\\\
\hat{x}^{2}\end{array}\\!\right)$ and
$\hat{A}=\left(\begin{array}[]{l}\hat{a}_{-}\\\
\hat{a}_{+}\end{array}\\!\right)$, then the ladder operators have the
following three important relations:

$\displaystyle[\hat{a}_{-},\hat{a}_{+}]$ $\displaystyle=$ $\displaystyle 1$
(3) $\displaystyle\left[\hat{L}^{2},\hat{a}_{\mp}\right]$ $\displaystyle=$
$\displaystyle\mp\,\lambda\,\hat{a}_{\mp}$ (4)
$\displaystyle\hat{A}^{\dagger}\hat{A}$ $\displaystyle=$
$\displaystyle\gamma\,\hat{X}^{\dagger}\hat{X}=\gamma\,\hat{L}^{2},$ (5)

where $\lambda$ and $\gamma$ are constants. If $\hat{A}=U\hat{X}$ denotes the
transformation from $\hat{X}$ to $\hat{A}$, then Eq.(5) implies that
$U^{\dagger}U=\gamma$. With $\theta^{12}=-\theta^{21}=\theta$ in 2-D, we can
define the ladder operators as

$\displaystyle\hat{a}_{\mp}=\frac{e^{\pm
i\delta}}{\sqrt{2\theta}}(\hat{x}^{1}\pm\,i\,\hat{x}^{2})$ (6)

that satisfy Eq.(3)-Eq.(5) with $\lambda=2\theta$ and $\gamma=1/\theta$. Here
$\delta$ is an arbitrary real number. The square of the length in this case is
given by

$\displaystyle\hat{L}^{2}=2\theta\left(\hat{a}_{+}\hat{a}_{-}+\frac{1}{2}\right).$
(7)

In analogy with the quantum harmonic oscillator problem, it is clear that on
the eigenstate $|n\rangle$ of $\hat{a}_{+}\hat{a}_{-}$,

$\displaystyle\hat{L}^{2}\,|n\rangle=2\theta[n+(1/2)]\,|n\rangle,$ (8)

with the minimum eigenvalue being $\theta$. The operator $\hat{a}_{+}$ plays
the role of a raising operation i.e., the eigenvalues of $\hat{L}^{2}$ on the
states $\left|n\right>$ and $\hat{a}_{+}\\!\left|n\right>$ are $2\theta$
apart. The operator $\hat{a}_{-}$ does the lowering operation. The angle
$\delta$ in Eq.(6) _corresponds_ to the orientation of the line segment that
would connect the points $y$ and $z$ in the commutative 2-D space.

On such admissible eigenstates, the square-root of Eq.(7) is also quantized.
In 2-D Euclidean space, the commutative analogue of Eq.(1) could also mean the
area of a square of side $L$ in which case the quantization of Eq.(7) can also
imply the quantization of area. In [25], it has been shown in the context of
spectral manifolds in NC geometry that the area of a 2-D manifold is
quantized. Here, we arrive at the same result in the case of 2-D plane with
coordinate noncommutativity. In the following, we attempt to generalize this
result to other NC spaces.

Before proceeding, note that the commutator Eq.(4) is essential to construct a
ladder of states which are all eigenstates of $\hat{L}^{2}$ operator and that
the canonical commutator Eq.(2) (or Eq.(3)) can be used to reduce the degree
of any polynomial $\hat{P}(\hat{x})$ by considering the commutator
$[\hat{P}(\hat{x}),\hat{x}^{i}]$. Eq.(4) is an example for such reduction of
the degree of a polynomial. In this context, it is worth remarking that a
generalized coordinate noncommutativity, which would involve a non-constant
$\theta^{ij}$ in Eq.(2), will pose problem to construct a ladder of
eigenstates of $\hat{L}^{2}$ using Eq.(4). So such generalized coordinate
commutator structure (see for example [4]) is not considered in this
work333The authors thank the referee for raising the issue of generalized
coordinate commutator structure..

##  2 A Generalization

It turns out that Eq.(3)-Eq.(5) play a crucial role in the generalization of
the idea of length quantization to other spaces with constant and symmetric
metric $g_{ij}$. In an $N$-dimensional space we define the lowering operator
and the operator corresponding to square of length respectively as

$\displaystyle\hat{a}_{-}=\alpha_{i}\,\hat{x}^{i},\qquad\hat{L}^{2}=g_{ij}\hat{x}^{i}\hat{x}^{j},\qquad
i=1,2,\ldots N,$ (9)

where Einstein’s summation convention is implied and $\alpha_{i}$’s are
complex constants to be determined. Substitution of Eq.(9) in Eq.(4) gives the
relation

$\displaystyle 2i\,g_{ij}\theta^{jk}\alpha_{k}=-\lambda\alpha_{i}.$ (10)

Also, Eq.(3) and Eq.(10) leads to

$\displaystyle\alpha^{i}\alpha_{i}^{*}=2/\lambda,$ (11)

where $\alpha_{i}^{*}$ is the complex conjugate of $\alpha_{i}$. In the
following, we solve Eq.(10) and Eq.(11) for the cases of 3-D space, 1+1
dimensional spacetime and 2+1 dimensional spacetime, and analyze the
consequences.

###  2.1 The Case of 3-D

We assume that $g_{ij}=\mathrm{diag}(1,1,1)$ and that
$\theta^{12}=\theta^{13}=\theta^{23}=\theta$, in which case the set of three
equations Eq.(10) have nontrivial solution only if the secular determinant

$\displaystyle\left|\begin{array}[]{ccc}\lambda&2i\theta&2i\theta\\\
-2i\theta&\lambda&2i\theta\\\
-2i\theta&-2i\theta&\lambda\end{array}\right|=0,$ (15)

which leads to the nontrivial value $\lambda=\pm 2\sqrt{3}\,\theta$. The third
trivial solution $\lambda=0$ leads to a set of $\alpha_{i}$’s such that
$\hat{a}_{+}\propto\hat{a}_{-}$, thereby violating Eq.(3). Also, $\lambda=0$
does not give any length-raising or lowering operation in the theory (see
Eq.(4)). Choosing the positive value for $\lambda$ so that $\hat{a}_{-}$ can
be identified with a lowering operation and substituting it in Eq.(10) and
Eq.(11) gives a set of values for $\alpha_{i}$’s:

$\displaystyle(\alpha_{1},\,\alpha_{2},\,\alpha_{3})=\left(\rho\sigma,\,-\rho\sigma^{*},\,\rho\right),$
(16)

where $\rho=\displaystyle e^{i\delta_{1}}/(\sqrt{3\theta\sqrt{3}})$ and
$\sigma=-e^{i\pi/3}$, and $\delta_{1}$ is a real constant. Note that
$|\alpha_{i}|=|\rho|=1/(\sqrt{3\theta\sqrt{3}})$. The lowering operator can
then be expressed as

$\displaystyle\hat{a}_{-}=\rho\left[\sigma\hat{x}^{1}-\sigma^{*}\hat{x}^{2}+\hat{x}^{3}\right].$
(17)

Since there are three independent operator variables
$\hat{x}^{1},\,\hat{x}^{2}$ and $\hat{x}^{3}$, we need another operator
$\hat{b}$ in addition to $\hat{a}_{-}$ and $\hat{a}_{+}$, to define the
transformation $\hat{A}=U\hat{X}$ in 3-D and to make it invertible. But
$\hat{b}$ and its Hermitian conjugate $\hat{b}^{\dagger}$ should not be
independent of each other, else we would end up with four independent operator
variables. We define in the following way:

$\displaystyle\hat{b}=\beta_{i}\,\hat{x}^{i},$ (18)

where $\beta_{i}$’s are complex numbers. Since $\hat{b}$ is defined as a
linear and homogeneous function of $\hat{x}^{i}$, its Hermitian conjugate
$\hat{b}^{\dagger}$ is also linear and homogeneous in $\hat{x}^{i}$ and so
$\beta_{i}^{*}=\beta\beta_{i}$, where $\beta$ is such that $|\beta|=1$ since
$|\beta_{i}^{*}|=|\beta_{i}|$. If
$\hat{A}^{\dagger}=(\hat{a}_{+},\,\hat{a}_{-},\,\hat{b}^{\dagger})$ and
$\hat{X}^{\dagger}=(\hat{x}^{1},\,\hat{x}^{2},\,\hat{x}^{3})$, then the analog
of Eq.(5) is written as

$\displaystyle\hat{L}^{2}=\hat{X}^{\dagger}g\hat{X}=\frac{1}{\gamma}\hat{A}^{\dagger}g\,\hat{A}=\frac{1}{\gamma}\hat{X}^{\dagger}U^{\dagger}g\,U\hat{X},$

where $g$ is the matrix form of the metric tensor. The above $\hat{L}^{2}$ can
further be expressed as

$\displaystyle\hat{L}^{2}=(\hat{x}^{1})^{2}+(\hat{x}^{2})^{2}+(\hat{x}^{3})^{2}=\frac{1}{\gamma}(2\hat{a}_{+}\hat{a}_{-}+\hat{b}^{\dagger}\hat{b}+1)$
(19)

upon the condition that $U^{\dagger}g\,U=\gamma\,g$. Using the expressions for
$\hat{a}_{-}$, $\hat{a}_{+}$ and $\hat{b}$ in terms of $\hat{x}^{i}$ as in
Eq.(9) and Eq.(18), we can get the elements of $U$ in terms of $\alpha_{i}$,
$\alpha_{i}^{*}$ and $\beta_{i}$ from which the condition
$U^{\dagger}g\,U=\gamma\,g$ in indicial form is expressed as

$\displaystyle\alpha_{i}^{*}\alpha_{j}+\alpha_{i}\alpha_{j}^{*}+\beta_{i}^{*}\beta_{j}=\gamma\,g_{ij}.$
(20)

For all the values $i=j$, the above condition gives $|\beta_{j}|$ as

$\displaystyle|\beta_{j}|=\sqrt{(\gamma-2|\rho|^{2})}.$ (21)

For $i=1$ and $j=2$ and $3$ we can respectively show that

$\displaystyle\beta_{2}=-\frac{1}{\beta_{1}^{*}}(\alpha_{1}^{*}\alpha_{2}+\alpha_{1}\alpha_{2}^{*}),\qquad\beta_{3}=-\frac{1}{\beta_{1}^{*}}(\alpha_{1}^{*}\alpha_{3}+\alpha_{1}\alpha_{3}^{*}),$
(22)

which in general leads to $\beta_{i}^{*}=\beta_{i}(\beta_{1}^{*}/\beta_{1})$.
Calculation of $|\beta_{1}^{*}\beta_{2}|$ using Eq.(22), (21) and (16) gives
$\gamma=3|\rho|^{2}=1/(\sqrt{3}\,\theta)$. Therefore, by writing
$\beta_{1}=|\rho|\,e^{i\delta_{2}}$, Eq.(22) completely determines $\beta_{2}$
and $\beta_{3}$ in terms of $\delta_{2}$ and $|\rho|$:

$\displaystyle(\beta_{1},\,\beta_{2},\,\beta_{3})=(|\rho|e^{i\delta_{2}},\,-|\rho|e^{i\delta_{2}},\,|\rho|e^{i\delta_{2}}).$
(23)

Then the only free parameters in the theory would be $\delta_{1}$ and
$\delta_{2}$. These two parameters would correspond to the orientation of the
line segment connecting the points $y$ and $z$ in the physical 3-D commutative
space.

Note that the explicit values Eq.(16) of the $\alpha_{i}$’s and $\beta_{i}$’s
have the following properties:

$\displaystyle\alpha_{i}\,\alpha^{i}$ $\displaystyle=$ $\displaystyle 0,$ (24)
$\displaystyle\theta^{ij}\beta_{j}$ $\displaystyle=$ $\displaystyle 0.$ (25)
$\displaystyle\alpha_{i}\,\beta^{i}$ $\displaystyle=$
$\displaystyle\alpha_{i}^{*}\,\beta^{i}=0,$ (26)

Eq.(24) can also be inferred from the $[\hat{a}_{-},\hat{a}_{-}]=0$. Eq.(26)
leads to $[\hat{a}_{-},\hat{b}]=[\hat{a}_{+},\hat{b}]=0$. In fact, Eq.(25)
leads to $[\hat{x}^{i},\hat{b}]=0$ and thus $\hat{b}$ commutes with all the
operators of the form $\hat{F}(\hat{x})$ in the theory. In other words, the
states characterized by any index corresponding to $\hat{b}$ are unaffected by
other operators of the form $\hat{F}(\hat{x})$.

Using Eq.(4), it is easy to show that
$[\hat{L}^{2},\hat{a}_{+}\hat{a}_{-}]=0$. So it is possible to construct a
complete set of simultaneous eigenstates of $\hat{L}^{2}$,
$\hat{a}_{+}\hat{a}_{-}$ and $\hat{b}^{\dagger}\hat{b}$.

Like in the case of quantum harmonic oscillator problem, the relations Eq.(3)
and Eq.(4) ensure that the eigenvalues of $\hat{a}_{+}\hat{a}_{-}$ should be
non-negative integers (since the eigenvalues of $\hat{b}^{\dagger}\hat{b}$
will also turn out to be non-negative), and the eigenstate itself may be
denoted by $|n\rangle$. Using the commutator

$\displaystyle[\hat{x}^{i},\,\hat{a}_{+}\hat{a}_{-}]=\frac{\lambda}{2}(\alpha^{i*}\,\hat{a}_{-}-\alpha^{i}\,\hat{a}^{\dagger}),$
(27)

we can show that

$\displaystyle\hat{a}_{+}\hat{a}_{-}(\hat{x}^{i}|n\rangle)=-\frac{\lambda}{2}\left(\alpha^{i*}\,\sqrt{n}|n-1\rangle-\alpha^{i}\,\sqrt{n+1}|n+1\rangle\right)+n(\hat{x}^{i}|n\rangle).$
(28)

The above relation suggests that the state $(\hat{x}^{i}|n\rangle)$ can be
written as

$\displaystyle\hat{x}^{i}|n\rangle=\chi_{(n)-1}^{i}|n-1\rangle+\chi_{(n)}^{i}|n\rangle+\chi_{(n)+1}^{i}|n+1\rangle,$
(29)

where $\chi_{(n)-1}^{i},~{}\chi_{(n)}^{i}$ and $\chi_{(n)+1}^{i}$ are
constants to be determined.

By acting $\hat{a}_{+}\hat{a}_{-}$ on Eq.(29) and comparing the result with
Eq.(28), we have

$\displaystyle\chi_{(n)-1}^{i}=\frac{\lambda}{2}\alpha^{i*}\sqrt{n}\,,\qquad\chi_{(n)+1}^{i}=\frac{\lambda}{2}\alpha^{i}\sqrt{n+1}$
(30)

Projecting the state Eq.(29) onto $|n\rangle$, $|n-1\rangle$ and $|n+1\rangle$
respectively and in those projections comparing the action of $\hat{x}^{i}$ on
the ket vector with its action on the bra vector lead to

$\chi_{(n)}^{i*}=\chi_{(n)}^{i},\qquad\chi_{(n-1)+1}^{i*}=\chi_{(n)-1}^{i},\qquad\chi_{(n+1)-1}^{i*}=\chi_{(n)+1}^{i}.$
(31)

Making use of the commutator
$[\hat{x}^{i},\,\hat{a}^{\dagger}]=\frac{\lambda}{2}\alpha^{i*}$ in the
calculation of $\langle n+1|a^{\dagger}\hat{x}^{i}|n\rangle$, we can show that
$\chi_{(n+1)}^{i}=\chi_{(n)}^{i}$ and therefore $\chi_{(n)}^{i}$ is
independent of $n$, which we denote by $\chi^{i}$. To be consistent with the
relations like $\hat{a}|n\rangle=\sqrt{n}|n-1\rangle$, $\chi^{i}$ needs to
satisfy

$\displaystyle\chi^{i}\,\alpha_{i}=0=\chi^{i}\alpha_{i}^{*},$ (32)

which is possible if we choose anyone of the following set for $\chi^{i}$:

$\displaystyle\chi^{i}=(+\chi,\,-\chi,\,+\chi)~{}~{}\mathrm{or}~{}~{}(-\chi,\,+\chi,\,-\chi),$
(33)

which gives $(\chi^{i})^{2}=\chi^{2}$ for any particular $i$, where $\chi$ is
some real constant. The values in Eq.(33) are motivated by the values of
$\beta^{i}$, Eq.(23), because both $\chi^{i}$ and $\beta^{i}$ have the same
properties with $\alpha_{i}$ (compare Eq.(32) and Eq.(26)).

If we choose the states
$\left(\frac{1}{\sqrt{\theta\xi^{i}_{n}}}\hat{x}^{i}|n\rangle\right)$ to
normalize to $1$, where $i$ is not summed over and $\xi^{i}_{n}$’s are
dimensionless constants to be determined, then using Eq.(29) and its Hermitian
conjugate, the normalization leads to

$\displaystyle|\xi_{n}^{i}|=\frac{1}{\theta}\left(\chi^{2}+\frac{\lambda^{2}|\rho|^{2}}{4}(2n+1)\right).$
(34)

We denote the above expression by $\xi_{n}(\chi)$ since it is the same for all
$i$ and it depends on $\chi$. Also, using Eq.(26), Eq.(30) and Eq.(33), it is
straightforward to show that

$\displaystyle\hat{b}|n\rangle$ $\displaystyle=$
$\displaystyle(\beta_{i}\chi^{i})|n\rangle=(3|\rho|\chi
e^{i\delta_{2}})|n\rangle,$ (35)
$\displaystyle\hat{b}^{\dagger}\hat{b}|n\rangle$ $\displaystyle=$
$\displaystyle(\beta_{i}^{*}\,\chi^{i})(\beta_{j}\,\chi^{j})|n\rangle=(\frac{\sqrt{3}\,\chi^{2}}{\theta})|n\rangle,$
(36)

and therefore the states may be properly denoted by $|n,\chi\rangle$ instead
of $|n\rangle$. Since the operator $\hat{b}$ commutes with all other
operators, it is not possible to determine $\chi$ through the operator algebra
method.

Finally, the eigenvalues of $\hat{L}^{2}$ in Eq.(19) are worked out to be

$\displaystyle\hat{L}^{2}|n,\chi\rangle=\theta\,\xi_{n}\,|n,\chi\rangle=(\chi^{2}+\frac{\theta}{\sqrt{3}}(2n+1))|n,\chi\rangle.$
(37)

Essentially, we have worked out the eigenvalues of $\hat{L}^{2}$ in terms of
the eigenvalues corresponding to $\hat{a}_{+}\hat{a}_{-}$ and
$\hat{b}^{\dagger}\hat{b}$. Since $\chi$ is real because of Eq.(31), the
minimum value for $\chi^{2}$ is $0$, and therefore we have,

$\displaystyle\hat{L}^{2}|n,0\rangle=\theta\,\xi_{n}\,|n,0\rangle=\frac{\theta}{\sqrt{3}}(2n+1)|n,0\rangle.$
(38)

In the Euclidean 3-D space, the commutative analogue of Eq.(19) can also be
interpreted as the area of a square of side $L$ in the plane formed by
$\hat{a}_{-}$ and $\hat{a}_{+}$, and so the quantization of Eq.(19) implies
the quantization of area with the minimum value being
$\frac{\theta}{\sqrt{3}}=\frac{2\theta^{\prime}}{\sqrt{3}}$. But the actual
uncertainty relation $(\Delta y^{1})(\Delta
y^{2})\geq\frac{\theta^{\prime}}{2}$ would yield the minimum
$\frac{\theta^{\prime}}{2}$ which is lower than
$\frac{2\theta^{\prime}}{\sqrt{3}}$. So the minimum value of the quantized
area is not violating the uncertainty principle. If a volume is written as a
function of the side $\hat{L}$, then the volume is also quantized along the
dimensions of $\hat{a}_{-}$ and $\hat{a}_{+}$.

###  2.2 1+1 Dimensional Spacetime

If $g_{ij}=\mathrm{diag}(1,-1)$ and $\theta^{12}=-\theta^{21}=\theta$, then
the relation Eq.(10) leads to the purely imaginary value for $\lambda=\pm
2i\theta$. Also, the substitution of this value back into Eq.(10) implies that
$\alpha_{1}=\mp\alpha_{2}$, which will contradict Eq.(3). Therefore, the
quantization of spacetime length in this approach is not feasible.

###  2.3 2+1 Dimensional Spacetime

If we take $g_{ij}=\mathrm{diag}(1,1,-1)$ and
$\theta^{12}=\theta^{13}=\theta^{23}=\theta$, then the solution to the secular
equation corresponding to Eq.(10) gives only the purely imaginary number
$\lambda=\pm 2i\theta$ as the nontrivial value. Putting the value
$\lambda=2i\theta$ in Eq.(10) gives no nontrivial solution to $\alpha_{i}$,
and the value $\lambda=-2i\theta$ results in
$\hat{a}^{\dagger}\propto\hat{a}$, contradicting Eq.(3). So, in this case
also, our method is not feasible to quantize the spacetime length. However, if
$\theta^{23}=0$, i.e., if the time commutes with the spatial coordinates, then
the method outlined in the Introduction can be used to quantize the spatial
part of $\hat{L}^{2}$ in a particular rectilinear system of coordinates.

###  2.4 3+1 Dimensional Spacetime

In this case, if the time is taken to commute with the spatial coordinates,
then the same method for the 3-D case can be used to quantize the spatial
length in a particular rectilinear system of coordinates. However, if time is
assumed to noncommute with spatial coordinates, it requires a separate
elaborate treatment and it will be published elsewhere.

##  3 Concluding Remarks

In this work, we have proposed a length operator $\hat{L}$ in Weyl-Moyal type
noncommutative (NC) spaces, and analyzed the consequences on the quantization
of length. By comparing the operator formalism of 2-D NC space with that of
quantum harmonic oscillator problem, we have deduced that the operator
corresponding to the square of length is analogous to the Hamiltonian of the
oscillator and hence the length and area are quantized in 2-D NC space. This
result is in conformity with and a special case of the already established
result in the context of spectral manifolds in NC geometry [25]. We have also
succeeded in showing that the length is quantized in a 2-D NC subspace of a
3-D NC space. Since the length quantization is more fundamental, the
quantization of area and volume [25] can be inferred from it for the special
cases in which they directly depend on the length along the 2-D subspace. But
our method does not work for the cases of 1+1 and 2+1 spacetime dimensions if
the time is taken to noncommute with the spatial coordinates. This is because
the eigenvalue equation Eq.(10) which is the result of the commutator Eq.(4)
gives imaginary eigenvalues for $\lambda$ — the quantum of $\hat{L}^{2}$. This
essentially implies that the length is increased or decreased in steps of
imaginary values which is unphysical. When the metric is changed to an
Euclidean metric, the quanta of $\hat{L}^{2}$ become real.

## Data Availability

No new data were created or analysed in this study.

## Conflicts of Interests

The authors declare that there is no conflict of interest regarding the
publication of this work.

## Funding Statement

The authors gratefully acknowledge the funding (No.PU/PS2/PHYS/Minor-
Equip/21-22/241) by Pondicherry University under the Minor Equipment Grant
No.PU/PD2/Minor-Equip/2022/536.

## Acknowledgment

A version of this work has been published as an arXiv preprint:
https://arxiv.org/abs/2206.07972 [27].

## References

  * [1] H. S. Snyder, Phys. Rev. 71, 38-41 (1947)
  * [2] T. Filk, Phys. Lett. B 376, 53-58 (1996)
  * [3] S. Doplicher, K. Fredenhagen and J. E. Roberts, Commun. Math. Phys. 172, 187-220 (1995) [arXiv:hep-th/0303037 [hep-th]].
  * [4] A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D 52 (1995), 1108-1118 doi:10.1103/PhysRevD.52.1108 [arXiv:hep-th/9412167 [hep-th]].
  * [5] N. Seiberg and E. Witten, JHEP 09, 032 (1999) [arXiv:hep-th/9908142 [hep-th]].
  * [6] J. Madore, [arXiv:gr-qc/9906059 [gr-qc]].
  * [7] P. Nicolini, Int. J. Mod. Phys. A 24, 1229-1308 (2009) [arXiv:0807.1939 [hep-th]].
  * [8] R. J. Szabo, Phys. Rept. 378, 207-299 (2003) [arXiv:hep-th/0109162 [hep-th]].
  * [9] B. Muthukumar, Phys. Rev. D 71, 105007 (2005) [arXiv:hep-th/0412069 [hep-th]].
  * [10] B. Muthukumar, JHEP 01, 123 (2015) [arXiv:1408.5478 [hep-th]].
  * [11] P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp and J. Wess, Class. Quant. Grav. 22, 3511-3532 (2005) [arXiv:hep-th/0504183 [hep-th]].
  * [12] X. Calmet and A. Kobakhidze, Phys. Rev. D 72, 045010 (2005) [arXiv:hep-th/0506157 [hep-th]].
  * [13] E. Harikumar and V. O. Rivelles, Class. Quant. Grav. 23, 7551-7560 (2006) [arXiv:hep-th/0607115 [hep-th]].
  * [14] A. P. Balachandran, T. R. Govindarajan, K. S. Gupta and S. Kurkcuoglu, Class. Quant. Grav. 23, 5799-5810 (2006) [arXiv:hep-th/0602265 [hep-th]].
  * [15] M. Roy and B. Muthukumar, [arXiv:2205.02479 [hep-th]].
  * [16] V. P. Nair and A. P. Polychronakos, Phys. Lett. B 505, 267-274 (2001) [arXiv:hep-th/0011172 [hep-th]].
  * [17] S. Bellucci, A. Nersessian and C. Sochichiu, Phys. Lett. B 522, 345-349 (2001) [arXiv:hep-th/0106138 [hep-th]].
  * [18] B. Muthukumar and P. Mitra, Phys. Rev. D 66, 027701 (2002) [arXiv:hep-th/0204149 [hep-th]].
  * [19] B. Muthukumar, JHEP 01, 073 (2007) [arXiv:hep-th/0609117 [hep-th]].
  * [20] B. Muthukumar, AIP Conf. Proc. 939, no.1, 359-362 (2007)
  * [21] S. Biswas, P. Nandi and B. Chakraborty, Phys. Rev. A 102, no.2, 022231 (2020) [arXiv:1911.03196 [hep-th]].
  * [22] K. Bolonek and P. Kosinski, Phys. Lett. B 547, 51-54 (2002) [arXiv:hep-th/0208162 [hep-th]].
  * [23] A. Muhuri, D. Sinha and S. Ghosh, Eur. Phys. J. Plus 136, no.1, 35 (2021) [arXiv:2006.16528 [quant-ph]].
  * [24] E. Harikumar, V. S. Kumar and A. Khare, Phys. Lett. B 589, 155-161 (2004) [arXiv:hep-th/0402064 [hep-th]].
  * [25] A. H. Chamseddine, A. Connes and V. Mukhanov, Phys. Rev. Lett. 114, no.9, 091302 (2015) [arXiv:1409.2471 [hep-th]].
  * [26] T. Thiemann, J. Math. Phys. 39, 3372-3392 (1998) [arXiv:gr-qc/9606092 [gr-qc]].
  * [27] B. Muthukumar and A. Rashid, [arXiv:2206.07972 [hep-th]].