# Chirality in Astrophysics

Axel Brandenburg  Nordita, KTH Royal Institute of Technology and Stockholm
University,
and Department of Astronomy, Stockholm University, SE-10691 Stockholm, Sweden
∗E-mail<EMAIL_ADDRESS>, Revision: 1.54
https://www.nordita.org/~brandenb

###### Abstract

Chirality, or handedness, enters astrophysics in three distinct ways. Magnetic
field and vortex lines tend to be helical and have a systematic twist in the
northern and southern hemispheres of a star or a galaxy. Helicity is here
driven by external factors. Chirality can also enter at the microphysical
level and can then be traced back to the parity-breaking weak force. Finally,
chirality can arise spontaneously, but this requires not only the presence of
an instability, but also the action of nonlinearity. Examples can be found
both in magnetohydrodynamics and in astrobiology, where homochirality among
biomolecules probably got established at the origin of life. In this review,
all three types of chirality production will be explored and compared.

###### keywords:

Magnetic helicity; chiral magnetic effect; gravitational waves; homochirality

##  1 Introduction

Chirality, or handedness, plays important roles in many different fields of
astrophysics, including astrobiology. There are three distinct types of
chirality production: (i) driven chirality due to external factors, (ii)
driven chirality due to intrinsic properties, and (iii) spontaneous chirality
production due to instability and nonlinearity. The primary applications for
these three types of chirality production are rather distinct, but one can
find unifying circumstances under which the different types can be
demonstrated and compared. One such circumstance is given by the presence of
magnetic fields.

Magnetic fields can experience twisting that makes them helical, but the
central question is what determines the sign of this twist—especially if it is
a systematic one, always being in the same sense. When there are extrinsic or
intrinsic factors such as the combination of rotation and stratification in a
star, or intrinsic factors such as the presence of fermions of one of two
handednesses, the answer is in principal clear. However, there can also be
spontaneous helicity production, where the sign depends ultimately on chance,
so both signs are possible under almost identical conditions. An example that
we discuss at the end of this review is in the field of magnetohydrodynamics
(MHD), where a magnetic field in a stratified atmosphere that exhibits a
magnetic buoyancy instability where, in the end, once nonlinearity plays a
role, one particular handedness dominates to nearly hundred percent. It is
this example that, in a broad sense, also carries over to astrobiology and the
origin of life, where one particular chirality of biomolecules eventually
dominates and leads, to what is known as homochirality. Magnetic fields are
probably not involved in the origin of life, but there are simple mathematical
analogies in both cases.

In Table 1, we summarize the three types of chirality production under the
“what” column or category. The “where” category lists some specific examples,
and the “how” category highlights some specific techniques for measuring
chirality for those three types.

Summary of three types of chirality production in astrophysics.  What Where
How Helicity, driven by stars, planets, magnetic field from Zeeman effect,
stratification & rotation galaxies polarization, in situ in solar wind Fermion
asymmetry, entire universe polarization patters, photon arrival axions
direction statistics, circularly polarized gravitational waves Spontaneous
chirality astrobiology, enantiospecific uptake of nutrients production MHD

We begin with some historical remarks highlighting the significance of
helicity of magnetic fields (Sect. 2). In Sects. 3–5, we discuss aspects of
the three types in more detail and then conclude in Sect. 6 with some
additional reflections. We emphasize that we use the terms chirality,
helicity, and handedness rather interchangeably, although technically this is
not always accurate.

##  2 Historical remarks

###  2.1 Kinetic helicity in fluid dynamics

In the context of fluid dynamics, the term helicity was coined by Moffatt,[1]
who identified the topological equivalence between the knottedness of vortex
lines in fluid dynamics and the kinetic helicity. In fact, the term kinetic
helicity was proposed by him in his 1969 paper on the grounds that this term
is also used in subatomic physics to describe the alignment or anti-alignment
of spin and momentum of fermions, for example. Mathematically, the mean
kinetic helicity density is defined as $\langle\bm{\omega}\cdot\bm{u}\rangle$,
where the vorticity $\bm{\omega}={\bm{\nabla}}\times\bm{u}$ is the curl of the
velocity $\bm{u}$. Kinetic helicity is a pseudoscalar, i.e., it changes its
sign when the system is inspected through a mirror. Likewise, $\bm{\omega}$ is
a pseudovector, so it is more meaningful to plot it with its sense of rotation
(which changes in a mirror), rather than a vector with an arrow at its end.

Figure 1: Illustration showing that the line integral $\oint_{\partial
S}\bm{A}\cdot{\rm d}{}\bm{\ell}$ along flux tube $\Phi_{1}$, can be written as
a surface integral over the enclosed surface $S$,
$\int_{S}({\bm{\nabla}}\times\bm{A})\cdot{\rm d}{}\bm{S}$, where $\partial S$
is the line along flux tube $\Phi_{1}$. However, the only nonvanishing
contribution comes from flux tube $\Phi_{2}$, i.e.,
$\int_{S_{2}}\bm{B}\cdot{\rm d}{}\bm{S}$.

###  2.2 Magnetic helicity conservation

The quantities that we shall focus on here are mainly the mean magnetic
helicity density $\langle\bm{A}\cdot\bm{B}\rangle$, where
$\bm{B}={\bm{\nabla}}\times\bm{A}$ is the magnetic field expressed in terms of
its vector potential $\bm{A}$, and the magnetic helicity over a volume
containing two interlocked flux loops,
$\int_{V_{1}+V_{2}}\bm{A}\cdot\bm{B}\,{\rm
d}{}V=\int_{V_{1}}\bm{A}\cdot\bm{B}\,{\rm
d}{}V+\int_{V_{2}}\bm{A}\cdot\bm{B}\,{\rm d}{}V$. The volumes $V_{1}$ and
$V_{2}$ are those of the two tubes. For $V_{1}$, we can split the volume
integral into a line integral $\oint_{\partial S}\bm{A}\cdot{\rm
d}{}\bm{\ell}$ along the flux tube and a surface integral
$\int_{S_{1}}\bm{B}\cdot{\rm d}{}\bm{S}$ across the tube, i.e.,

$\int_{V_{1}}\bm{A}\cdot\bm{B}\,{\rm d}{}V=\left(\int_{S_{1}}\bm{B}\cdot{\rm
d}{}\bm{S}\right)\left(\oint_{\partial S}\bm{A}\cdot{\rm
d}{}\bm{\ell}\right);$ (1)

see Figure 1. Using Stokes’ theorem, the line integral along tube 1 can be
rewritten as a surface integral over the magnetic field going through the flux
ring, but its only nonvanishing contribution comes from the other intersecting
tube with surface $S_{2}$. Thus, we have $\int_{V_{1}}\bm{A}\cdot\bm{B}\,{\rm
d}{}V=\pm\Phi_{1}\Phi_{2}$, and likewise for $V_{2}$, so we get a factor 2.
The sign depends on the relative orientation of the field vectors in the two
tubes, so we write here $\int\bm{A}\cdot\bm{B}\,{\rm d}{}V=\pm
2\Phi_{1}\Phi_{2}$.

The magnetic field is a pseudovector. Its evolution is governed by the
homogeneous Maxwell equation $\partial\bm{B}/\partial t=-\nabla\times\bm{E}$,
where $t$ is time and $\bm{E}$ is the electric field. Since
${\bm{\nabla}}\cdot\bm{B}=0$, it is convenient to consider the uncurled
induction equation

$\partial\bm{A}/\partial t=-\bm{E}-{\bm{\nabla}}\phi,$ (2)

where $\phi$ is the electric (or scalar) potential. To obtain an evolution
equation for $\bm{A}\cdot\bm{B}$, we compute
$\bm{A}\cdot\dot{\bm{B}}+\dot{\bm{A}}\cdot\bm{B}$, where dots denote partial
time differentiation, and find

$\frac{\partial}{\partial
t}\left(\bm{A}\cdot\bm{B}\right)=-2\bm{E}\cdot\bm{B}-{\bm{\nabla}}\cdot(\underbrace{\phi\bm{B}+\bm{E}\times\bm{A}}_{\mbox{helicity
flux}}).$ (3)

We see that the magnetic helicity production depends on $\bm{E}\cdot\bm{B}$
and on the presence of magnetic helicity fluxes, $\mbox{\boldmath${\cal
F}$}{}_{H}=\phi\bm{B}+\bm{E}\times\bm{A}$. The $\bm{E}\cdot\bm{B}$ term plays
important roles in electrically non-conducting environments, for example
during inflation in the early universe when space-time became extremely
diluted. By contrast, in the contemporary universe, and even in the space
between galaxy clusters, there is still sufficient conductivity so that the
laws of MHD apply, and not the vacuum equations for electromagnetic waves,
which apply during inflation. In MHD, the electric field is given by
$\bm{E}=-\bm{u}\times\bm{B}+\bm{J}/\sigma$, so the first term does not
contribute to $\bm{E}\cdot\bm{B}$. Thus, the only contribution comes from
$\bm{J}\cdot\bm{B}/\sigma$, so we have

$\frac{\partial}{\partial
t}\langle\bm{A}\cdot\bm{B}\rangle=-2\eta\langle\bm{J}\cdot\bm{B}\rangle,$ (4)

where $\eta=(\mu_{0}\sigma)^{-1}$ is the magnetic diffusivity, and angle
brackets denote volume averaging. When the conductivity is large, $\eta$ is
small, and $\eta\langle\bm{J}\cdot\bm{B}\rangle$ converges to zero like
$\eta^{1/2}$ as $\eta\to 0$.

The smallness of $\eta$ in many astrophysical settings makes
$\langle\bm{A}\cdot\bm{B}\rangle$ nearly perfectly conserved,[2] except for
the presence of magnetic helicity fluxes. Those vanish in homogeneous systems
(e.g., in homogeneous helical turbulence), but astrophysical dynamos are
usually not homogeneous and magnetic helicity fluxes, for example out of the
star or between its northern and southern hemispheres, are believed to play
important roles in astrophysical dynamos to alleviate some serious
constraints[3] arising from the magnetic helicity conservation otherwise.

###  2.3 Helicity-driven large-scale dynamos

To understand the aforementioned constraint, we need to emphasize that
magnetic helicity is usually connected with the presence of kinetic helicity,
$\langle\bm{\omega}\cdot\bm{u}\rangle$. It can lead to an electromotive force
along the mean magnetic field, ${\overline{\bm{B}}}$, of the form

$\overline{\bm{u}\times\bm{b}}=\alpha{\overline{\bm{B}}}-\eta_{\rm
t}\mu_{0}{\overline{\bm{J}}},$ (5)

where $\bm{u}=\bm{U}-\overline{\bm{U}}$ and
$\bm{b}=\bm{B}-{\overline{\bm{B}}}$ are fluctuations of velocity and magnetic
fields, and
${\overline{\bm{J}}}={\bm{\nabla}}\times{\overline{\bm{B}}}/\mu_{0}$ is the
mean current density. Here, overbars denote averaging (for example planar or
$xy$ averaging), but the type of averaging depends on the particular problem.
Under the assumption of isotropy and high conductivity,[4] we have
$\alpha=-\langle\bm{\omega}\cdot\bm{u}\rangle\tau/3$ and $\eta_{\rm
t}=\langle\bm{u}^{2}\rangle\tau/3$, and $\tau$ is the correlation time. The
$\alpha{\overline{\bm{B}}}$ term can lead to an exponential growth of
${\overline{\bm{B}}}$. The second term just leads to an enhancement of the
microphysical magnetic diffusion, $\eta\mu_{0}{\overline{\bm{J}}}$.

The $\alpha$ effect leads to the generation of magnetic helicity of the mean
field with
${\overline{\bm{A}}}\cdot{\overline{\bm{B}}}=O({\overline{\bm{B}}}^{2}/k_{\rm
m})\neq 0$, where $k_{\rm m}$ is the typical wavenumber of the magnetic field.
However, since the total magnetic helicity is conserved (and vanishing of the
field was very small initially), we must generate small-scale magnetic
helicity, $\langle\bm{a}\cdot\bm{b}\rangle$, of opposite sign. The resulting
current helicity, $\langle\bm{j}\cdot\bm{b}\rangle\approx k_{\rm
f}^{2}\langle\bm{a}\cdot\bm{b}\rangle$, with some typical wavenumber $k_{\rm
f}$ characterizing the fluctuation scales, quenches the $\alpha$ effect and
leads to slow dynamo saturation.[5]

We will not go into further details here, but refer the reader to reviews on
the subject.[3, 6] This field of research is still very active and there are
still important questions regarding magnetic helicity fluxes that remain
controversial.

##  3 Chirality driven by external factors

###  3.1 Examples of helicities: even and odd in $\bm{B}$

We discuss here three different helicities: kinetic helicity
$\langle\bm{\omega}\cdot\bm{u}\rangle$, magnetic helicity
$\langle\bm{A}\cdot\bm{B}\rangle$, and cross helicity
$\langle\bm{u}\cdot\bm{B}\rangle$. The latter reflects the linkage between an
$\bm{\omega}$ tube (vortex tube) and a $\bm{B}$ tube (magnetic flux tube). To
understand the production of kinetic helicity, one has to realize that it is a
pseudoscalar. This means, that it is the product of a polar vector and an
axial vector. Rotation, for example, is an axial vector, but that alone cannot
produce magnetic or kinetic helicity. However, in the presence of both
rotation $\bm{\Omega}$ and gravitational stratification characterized by the
gravitational acceleration $\bm{g}$, which is a polar vector, we can produce
the pseudoscalar $\bm{g}\cdot\bm{\Omega}$. Thus, kinetic helicity can be
produced if one can identify external factors such as the combined presence of
$\bm{g}$ and $\bm{\Omega}$ that could explain the presence of a non-vanishing
helicity. However, these external factors must also be even in the magnetic
field, so a nonvanishing $\langle\bm{u}\cdot\bm{B}\rangle$ with a systematic
sign cannot be explained in that way. Finite cross helicity can, however, be
explained if rotation was replaced by an imposed magnetic field $\bm{B}_{0}$,
so that $\bm{g}\cdot\bm{B}_{0}$ would be finite.

We thus see that $\langle\bm{\omega}\cdot\bm{u}\rangle$ may be linked to
$\bm{g}\cdot\bm{\Omega}$, although this needs to be (and has been) verified
using a detailed calculation.[4] In spherical coordinates $(r,\theta,\phi)$,
where $\theta$ is colatitude and the unit vectors of $\bm{g}$ and
$\bm{\Omega}$ are $\hat{\bm{g}}=(-1,0,0)$ and
$\hat{\bm{\Omega}}=(\cos\theta,-\sin\theta,0)$, respectively, we have
$\bm{g}\cdot\bm{\Omega}=-\cos\theta$, which is negative in the north and
positive in the south. This is indeed consistent with the observed sign of
$\langle\bm{\omega}\cdot\bm{u}\rangle$, and it is also found to govern the
sign of the magnetic helicity, but only at small and moderate length scales,
i.e., $\langle\bm{a}\cdot\bm{b}\rangle$.

Regarding the cross helicity, there is indeed a systematic large-scale
magnetic field $\bm{B}_{0}$ at the solar surface, although there can be
several sign changes in each hemisphere. On smaller scales of $\geq 20\,{\rm
Mm}$, it also changes between the two sides of a sunspot pair, which is
consistent with observations.[7] It turns out that
$\langle\bm{u}\cdot\bm{B}\rangle\approx-(\eta_{\rm t}/c_{\rm
s}^{2})\,\bm{g}\cdot\bm{B}_{0}$, where $\eta_{\rm t}$ is the turbulent
magnetic diffusivity and $c_{\rm s}$ is the sound speed.[8] This production
mechanism of cross helicity may play a role in theories of shallow sunspot
formation.[9, 10]

###  3.2 Observing helicity

####  3.2.1 Measuring magnetic helicity from solar magnetograms

Above the surface of the Sun, one often sees twisted structures in extreme
ultraviolet and x-ray images, which are suggestive of helicity.[11, 12, 13]
Even space observations of the surface of the Earth reveal cyclonic cloud
patterns that have opposite orientation in the northern and southern
hemispheres. However, to draw a connection with magnetic helicity, one must
first detect the magnetic field. This is possible through the Zeeman effect,
which causes circular polarization proportional to the line-of-sight magnetic
field, and linear polarization related to the perpendicular magnetic field
component – except for a $\pi$ ambiguity. For strong enough magnetic fields, a
“disambiguation procedure” based on a minimal magnetic energy assumption
allows one to determine the full $\bm{B}$ vector at the solar surface.[14, 15]
However, there is still not enough information about the changes of
polarization parameters along the line of sight, i.e., below and above the
surface of the Sun.

To make progress, one has to make some extra assumptions. One possibility is
to determine just the components normal to the surface, $B_{\|}\equiv B_{z}$
and $J_{\|}\equiv J_{z}=\partial_{x}B_{y}-\partial_{y}B_{x}$, where local
Cartesian coordinates $(x,y,z)$ have been employed, and compute
$J_{\|}B_{\|}$. This was first done by Seehafer,[16] who found that
$J_{\|}B_{\|}$ is negative in most of the active regions in the northern
hemisphere of the Sun, and positive in most of the active regions in the
southern one. From $J_{\|}$ and $B_{\|}$, one can also determine a proxy of
the magnetic helicity spectra,
$\tilde{A}_{\|}(\bm{k}_{\perp})\tilde{B}^{*}_{\|}(\bm{k}_{\perp})$, where
tildes denote Fourier transformation, $\bm{k}_{\perp}$ is the wavevector in
the horizontal ($xy$ surface) plane, and the asterisk denotes complex
conjugation. Here, $\tilde{A}_{\|}=\tilde{J}_{\|}/\bm{k}_{\perp}^{2}$. This
has been done[17, 18] and magnetic energy and helicity spectra are shown in
Figure 2(a) for active region AR 11158. Subsequent work[7] also presented
cross helicity spectra.

####  3.2.2 Magnetic helicity spectra from a time series

A completely different approach, that also makes use of Fourier
transformation, is to use in situ observations of time series of the $\bm{B}$
vector in the solar wind at one point in space. One can then make use of what
is known as the Taylor hypothesis to associate the temporal changes with
different positions through $\bm{r}=\bm{r}_{0}-\mbox{\boldmath$v$}{}t$, where
$v$ is the velocity vector of the solar wind, and $\bm{r}_{0}$ is a reference
position. Assuming homogeneity, one can write the magnetic two-point
correlation tensor in Fourier space as

$4\pi\langle\tilde{B}_{i}(\bm{k})\tilde{B}_{j}(\bm{k})\rangle=(\delta_{ij}-\hat{k}_{i}\hat{k}_{j})2\mu_{0}E_{\rm
M}(k)-{\rm i}\epsilon_{ijk}\hat{k}_{k}H_{\rm M}(k),$ (6)

where $E_{\rm M}(k)$ is the magnetic energy spectrum, normalized such that
$\int E_{\rm M}(k)\,{\rm d}{}k=\langle\bm{B}^{2}\rangle/2\mu_{0}$, and $H_{\rm
M}(k)$ is the magnetic helicity spectrum with $\int H_{\rm M}(k)\,{\rm
d}{}k=\langle\bm{A}\cdot\bm{B}\rangle$. This procedure revealed a clear
hemispheric antisymmetry with respect to the solar equator.[19]

Figure 2:  Magnetic energy and magnetic helicity spectra for southern
latitudes (a) at the solar surface in active region AR 11158, and (b) in the
solar wind at $\sim 1\,{\rm AU}$ distance. Positive (negative) signs are shown
as red open (blue filled) symbols. Positive signs are the solar surface at
intermediate and large $k$ correspond to positive values in the solar wind at
small $k$.

Comparing Figure 2(a) and (b), which are here both for the southern
hemisphere, we see that at the solar surface, $H_{\rm M}(k)$ has the expected
sign at small and intermediate scales ($k>0.1\,{\rm Mm}^{-1}$). In the solar
wind, however, a positive sign is only seen at very large scales ($k<30\,{\rm
AU}^{-1}$). The typical wavenumber at which the spectral helicity changes sign
at the solar surface of radius $700\,{\rm Mm}$ is $k\approx 0.1\,{\rm
Mm}^{-1}$. Expanding this linearly to a distance of $1\,{\rm AU}$ yields a
corresponding wavenumber of $k=70\,{\rm AU}^{-1}$, which is indeed where the
spectrum in the solar wind changes sign; see Figure 2(b). However, the change
is here the other way around: from positive to negative at large $k$. The
reason for this sign mismatch is not yet fully understood, but it is worth
noting that a sign reversal has also been seen in idealized numerical
simulations of stellar and galactic dynamos embedded in a turbulent
exterior.[20, 21, 22]

####  3.2.3 Canceling Faraday depolarization with helicity

There is an intriguing possibility to determine magnetic helicity from the
cancelation of Faraday depolarization.[23, 24] It might be particularly
suitable for calculating the magnetic helicity in the outskirts of edge-on
galaxies, where one might see the sign of magnetic helicity being reversed.

Normally, in the absence of magnetic helicity, Faraday rotation causes Faraday
depolarization. This is caused by the superposition of different polarization
planes from different depths along the line of sight. At the same time,
however, the perpendicular component of the magnetic field itself can rotate
about the line of sign if it is helical. These two effects can then either
enhance each other (and make the Faraday depolarization more complete), or
they can offset each other and lead to increased transmission or reduced
depolarization.

Mathematically, this can be seen by considering the observable polarization,
written in complex form as

$P(x,z,\lambda^{2})\equiv Q+{\rm
i}U=p_{0}\int_{-\infty}^{\infty}\epsilon\,e^{2{\rm
i}(\psi_{P}+{\phi}\lambda^{2})}\,{\rm d}{}y,$ (7)

where $\psi_{P}=\psi_{B}+\pi/2$ is the electric field angle, $\psi_{B}={\rm
atan}(B_{y},B_{x})$ is the magnetic field angle, $\epsilon(x,y,z)$ is the
emissivity, $p_{0}$ is the degree of polarization, $\lambda$ is the
wavelength,

$\phi(x,y,z)=-K\int_{-\infty}^{y}n_{\rm
e}(x,y^{\prime},z)\,B_{\|}(x,y^{\prime},z)\,{\rm d}{}y^{\prime}$ (8)

is the Faraday depth, with $n_{\rm e}$ being the electron density and
$K=0.81\,{\rm m}^{-2}\,{\rm cm}^{3}\,\mu{\rm G}^{-1}\,{\rm pc}^{-1}=2.6\times
10^{-17}\,{\rm G}^{-1}$ being a constant. Evidently, Faraday depolarization is
canceled if $\psi_{P}+{\phi}\lambda^{2}=0$.

In essence, for edge-on galaxies, we expect maximum polarized emission in
diagonally opposite quadrants of a galaxy; see Fig. 19.12 of Ref. 25. This
technique has also been applied to synthetic data of the solar corona.[26]

####  3.2.4 Magnetic helicity proxy

In the context of cosmology, a proxy of parity breaking and finite helicity of
the cosmic microwave background and the Galactic foreground is obtained by
decomposing the observed linear polarization in the sky into parity-even and
parity-odd contributions. This is done by expanding the complex polarization
$P\equiv Q+{\rm i}U$ with Stokes parameters $Q$ and $U$ into a spin-2
spherical harmonics as[27, 28]

$\tilde{R}_{\ell m}=\int_{4\pi}(Q+{\rm i}U)\,_{2}Y_{\ell
m}^{\ast}(\theta,\phi)\,\sin\theta\,{\rm d}{}\theta\,{\rm d}{}\phi.$ (9)

The parity-even ($E$) and parity-odd ($B$) contributions are obtained as the
real and imaginary parts of the return transformation as[29, 30]

$E+{\rm i}B\equiv
R=\sum_{\ell=2}^{N_{\ell}}\sum_{m=-\ell}^{\ell}\tilde{R}_{\ell m}Y_{\ell
m}(\theta,\phi).$ (10)

In cosmology, one usually considers correlations between $E$ and $B$, as well
as temperature and $B$, for example. However, it is also useful to consider
$B$ on its own, as was done in the context of the solar magnetic field[31] and
in the context of the Galactic magnetic field.[32] The result for our Galaxy
is shown in Figure 3.

Figure 3: Left: Galactic $B$ mode polarization. Right: longitudinally averaged
$B$ mode polarization. Here, $\theta$ and $\phi$ are Galactic colatitude and
longitude.

We see that the $B$ polarization is locally antisymmetric about the equatorial
plane, but there are canceling contributions from different longitudes, so the
longitudinal average is much smaller. Nevertheless, even the longitudinal
average does show a hemispheric antisymmetry.

Comparing with synthetic observations using the magnetic field from idealized
dynamo models[33] shows that this hemispheric dependence does not originate
from the different signs of the magnetic helicity expected in the northern and
southern hemispheres, but from the spiral pattern of our Galaxy, where the
views from the north and south correspond to mirror images of each other.

##  4 Magnetic helicity throughout the whole Universe

There is at present no definitive observation of finite helicity throughout
all of the Universe, but the possibility certainly exists.[34] In this
section, we first discuss two quite different mechanisms. One is related to
the chiral magnetic effect (CME) and the other to inflationary magnetogenesis.
Both generate helical magnetic fields, and the electromagnetic stress can
generate relic gravitational waves (GWs) that are circularly polarized. Such
waves, once generated, would not dissipate and would only dilute under the
cosmic expansion. They could still be observable with space
interferometers[35, 36, 37] and with pulsar timing arrays.[38] Measuring
circular polarization could provide a clean mechanism for determining the sign
of helicity in the Universe.

In the following, we describe the two generation mechanisms and compare in
Figure 4 the growth of the resulting magnetic field and the circular
polarization of GWs that could be observed in future using space
interferometers in the millihertz range.[35, 36, 37]

###  4.1 The CME

The CME is a quantum effect associated with the systematic alignment of the
spin $s$ of electrically charged fermions (electrons or positrons, for
example) with the momentum $\bm{p}$. A nonvanishing net helicity
$\mbox{\boldmath$s$}{}\cdot\bm{p}$ originates from the parity-breaking weak
force and manifests itself in the $\beta$ decay, for example, where spin and
momentum of electrons are antialigned, i.e.,
$\mbox{\boldmath$s$}{}\cdot\bm{p}<0$. The sign would be opposite for
antimatter, i.e., for positrons, for example. However, at low energies, spin
flipping occurs,[39] making this effect important only for highly relativistic
plasmas at high enough temperatures.

Is the presence of a magnetic field, the spin of chiral fermions aligns itself
with the magnetic field, and, owing to the finite momentum and charge of the
fermions, they produce a net current along the magnetic field,[40, 41, 42, 43,
44]

$\bm{J}=24\alpha_{\rm em}\,(n_{\rm L}-n_{\rm R})\left(\frac{k_{\rm B}T}{\hbar
c}\right)^{2}\bm{B},$ (11)

where $\alpha_{\rm em}\approx 1/137$ is the fine structure constant, $n_{\rm
L}$ and $n_{\rm R}$ are the number densities of left- and right-handed
fermions, respectively, $k_{\rm B}$ is the Boltzmann constant, $T$ is the
temperature of the Universe today, $\hbar$ is the reduced Planck constant, and
$c$ is the speed of light.

We see the analogy with Eq. (5), where we had a mean electromotive force
$\overline{\mbox{\boldmath${\cal E}$}}{}\equiv\overline{\bm{u}\times\bm{b}}$
with a mean current ${\overline{\bm{J}}}=(\alpha/\eta_{\rm
t}\mu_{0})\,{\overline{\bm{B}}}$ produced along the mean magnetic field for
turbulent for helical turbulence. Similar to the dynamo effect in helical
turbulence, there is a dynamo effect associated with chiral plasmas. This was
discussed in the context of cosmology[41] as a mechanism for producing
magnetic fields in the early Universe. This effect produces helical magnetic
fields, and the associated helicity reduces the net chirality of the fermions
such that the total chirality is conserved, i.e.,

$(n_{\rm L}-n_{\rm R})+\frac{4\alpha_{\rm em}}{\hbar
c}\langle\bm{A}\cdot\bm{B}\rangle={\rm const}{}.$ (12)

The length scales are small compared with the Hubble radius at any given time
and the resulting field strengths are limited to[45]

$|\langle\bm{A}\cdot\bm{B}\rangle|\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr<\cr\sim\cr}}}}\xi_{\rm
M}\langle\bm{B}^{2}\rangle\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr<\cr\sim\cr}}}}(0.5\times
10^{-18}\,{\rm G})^{2}\,{\rm Mpc},$ (13)

where $\xi_{\rm M}$ is the magnetic correlation length. This value of
$\xi_{\rm M}\langle\bm{B}^{2}\rangle$ is below the lower limits derived from
the non-observation of secondary GeV photons that are expected from inverse
Compton scattering of TeV photons from energetic blazers. One can therefore
not be very hopeful that those lower limits on the magnetic field can be
explained as a result of the CME. However, stronger fields could still be
produced during inflation, as will be discussed next.

###  4.2 Inflationary magnetogenesis

Models of inflationary magnetogenesis tend to invoke conformal invariance-
breaking to generate magnetic fields as a result of stretching without
diluting the magnetic energy during the inflationary expansion and the
subsequent reheating phase when most of the relevant particles and photons
were produced. Conformal invariance-breaking means that the term
$F_{\mu\nu}F^{\mu\nu}$ in the Lagrangian density is replaced by
$f^{2}F_{\mu\nu}F^{\mu\nu}$ with $f\neq 1$ during inflation[46] and
reheating.[47] In the presence of such a term, the vacuum evolution equation
for the vector potential changes from a standard wave equation
$(\partial^{2}/\partial t^{2}+k^{2})\tilde{\mathbf{A}}=0$ to
$(\partial^{2}/\partial
t^{2}+k^{2}-f^{\prime\prime}/f)(f\tilde{\mathbf{A}})=0$, where $t$ is now
conformal time and the primes on $f$ denote derivatives with respect to
conformal time.

Figure 4: (a) Growth of magnetic field from the CME (orange) and inflationary
magnetogenesis (red), where time is here expressed in terms of the scale
factor of the Universe $a(t)$, which increases monotonically. (b) Circular
polarization spectra of GWs produced from the CME (orange) and inflationary
magnetogenesis (red).

Commonly adopted forms of $f$ include $f\propto a^{\alpha}$ during inflation
and $f\propto a^{-\beta}$ during reheating.[48, 49, 50, 51]. In the presence
of pseudoscalars $\gamma$, such as axions, one also expects terms of the form
$\gamma f^{2}F_{\mu\nu}\tilde{F}^{\mu\nu}$ in the Lagrangian density. The
evolution equation for $\tilde{\mathbf{A}}$ takes then the form

$\left(\frac{\partial^{2}}{\partial t^{2}}+k^{2}\pm 2\gamma
k\frac{f^{\prime}}{f}-\frac{f^{\prime\prime}}{f}\right)(f\tilde{A}_{\pm})=0,$
(14)

where
$\tilde{\mathbf{A}}=\tilde{A}_{+}\hat{\mbox{\boldmath$e$}}{}_{+}+\tilde{A}_{-}\hat{\mbox{\boldmath$e$}}{}_{-}$
has been expressed in terms of the polarization basis
$\tilde{\mathbf{e}}_{\pm}(\bm{k})=[\tilde{\mathbf{e}}_{1}(\bm{k})\pm{\rm
i}\tilde{\mathbf{e}}_{2}(\bm{k})]/\sqrt{2}\,{\rm i}$ with ${\rm
i}\bm{k}\times\tilde{\mathbf{e}}_{\pm}=\pm k\tilde{\mathbf{e}}_{\pm}$, and
$\tilde{\mathbf{e}}_{1}(\bm{k})$, $\tilde{\mathbf{e}}_{2}(\bm{k})$ represent
units vectors orthogonal to $\bm{k}$ and orthogonal to each other. Figure 4(a)
shows the resulting algebraic growth of the magnetic field in comparison with
the exponential growth from the CME, and panel (b) shows the circular
polarization spectrum ${\cal P}_{\rm GW}(k)$. It is defined as[52]

$\\!\\!{\cal P}_{\rm GW}(k)=\\!\left.\int\\!2\,\mbox{\rm
Im}\,\tilde{h}_{+}\tilde{h}_{\times}^{*}\,k^{2}{\rm
d}{}\Omega_{k}\right/\\!\\!\int\\!\left(|\tilde{h}_{+}|^{2}+\tilde{h}_{\times}|^{2}\right)k^{2}{\rm
d}{}\Omega_{k}.$ (15)

We see that the degree of polarization reaches nearly 100% in a certain range.
[53, 54]

###  4.3 Detecting handedness from unit vectors in the sky

In addition to measuring the polarization of GWs as an indicator of the
helicity of the underlying magnetic field, there is yet another interesting
method that we describe here briefly. Suppose we observed energetic photons
from a particular astrophysical source from three slightly different
directions, $\hat{\bm{n}}_{1}$, $\hat{\bm{n}}_{2}$, and $\hat{\bm{n}}_{3}$,
and that their energies $E_{i}$ are ordered such that $E_{1}<E_{2}<E_{3}$,
then we can construct a pseudoscalar

$Q=(\hat{\bm{n}}_{1}\times\hat{\bm{n}}_{2})\times\hat{\bm{n}}_{3}.$ (16)

The quantity $Q$ would change sign in a mirror image of our Universe. The
value of $Q$ may well be zero within error bars, but if it is not, its sign
must mean something.

Detailed calculations have shown the TeV photons from blazars (i.e., the
accretion disk around supermassive black holes) can upscatter on the cosmic
background light and produce GeV photons.[55] Using about 10,000 photons
observed with Fermi Large Area Telescope data over a period of about five
years, Tashiro et al.[56] found $Q<0$ for all possible photon triples in
certain energy ranges. They interpreted this as evidence in favor of a
baryogenesis scenario that proceeds through changes in the Chern-Simons
number, which implies the generation of magnetic fields of negative
helicity.[57]

In all the studies of $Q$ done since then,[58, 59, 60, 61] one introduced a
cutoff toward low Galactic latitudes so as to avoid excess contamination from
our Galaxy. Using synthetic data, it has been found[62] that it is this
procedure that leads to the occurrence of large statistical errors in the
estimate of $Q$. Updated observations covering 11 years turned out to be no
longer compatible with a detection of a negative value of $Q$.[62] This has
been confirmed in subsequent work.[63] Thus, although this method could work
in principle, it would require much better statistics.

In this connection, it is interesting to note that an equivalent quantity $Q$
can be determined for a variety of different observations. Suppose there are
three sunspots of different strengths on the surface of the Sun, this again
implies a finite handedness. We can then ask whether this handedness could be
linked to the helicity of the underlying magnetic field. Idealized model
calculations have shown that this is indeed the case.[64]

##  5 Spontaneous chirality production

###  5.1 Biological homochirality

In astrobiology, an important question concerns the origin of biological
homochirality.[65] In solution, many organic molecules tend to rotate the
polarization plane of linearly polarized light. One refers to the substances
as either levorotatory (l for left-handed) or dextrorotatory (d for right-
handed). Almost all amino acids of terrestrial life are of the l form, and
almost all sugars are of the d form, for example the sugars in the phosphorus
backbones of deoxyribose nucleic acid (DNA). The origin of this homochirality
can be explained in terms of two essential processes: autocatalysis and mutual
antagonism – an old idea that goes back to a paper by F. C. Frank[66] of 1953.
Interestingly, this is the same year when Watson and Crick[67] discovered the
helix structure of DNA.

Autocatalysis produces “more of itself”, i.e., it can catalyze the formation
of chiral molecules of the l form from an achiral substrate $A$ in the
presence of l and, conversely, it can catalyze the formation of molecules of
the d form in the presence of d. The corresponding reactions

$L+A\to 2L\quad\mbox{and}\quad D+A\to 2D,$ (17)

with rate coefficient $k_{\rm C}$, imply that the associated concentrations
$[L]$ and $[D]$ obey

$\frac{{\rm d}{}}{{\rm d}{}t}[L]=k_{\rm
C}[A][L]-...\quad\mbox{and}\quad\frac{{\rm d}{}}{{\rm d}{}t}[D]=k_{\rm
C}[A][D]-...,$ (18)

which leads to exponential growth with time $t$ of both $[L]=[L]_{0}e^{k_{\rm
C}[A]t}$ and $[D]=[D]_{0}e^{k_{\rm C}[A]t}$, with initial values $[L]_{0}$ and
$[D]_{0}$. However, this process alone does not change the enantiomeric
excess, $\mbox{e.e.}=([L]-[D])/([L]+[D])$, which will always be equal to the
initial value. This is because we still need mutual antagonism, which will be
explained next.

For a long time, it remained unclear what would correspond to Frank’s mutual
antagonism. The relevant understanding was put forward by Sandars.[68] At that
time, it was thought that homochirality was a prerequisite to the origin of
life. This idea was based on an experimental result by Joyce et al.,[69] which
showed that in template-directed polymerization of oligomers of one chirality,
polymers of the opposite chirality terminate further polymerization. This was
called enantiomeric cross-inhibition, and was regarded as a serious problem
for the origin of life, and that life could only emerge in a fully homochiral
environment.[70] Sandars realized that enantiomeric cross-inhibition could
just be the crucial mechanism that corresponds to Frank’s mutual antagonism
that would lead to the emergence of homochirality. The corresponding reaction
and rate equations, with rate coefficient $k_{\rm I}$, are

$L+D\to 2A\quad\mbox{and}\quad\frac{{\rm d}{}}{{\rm d}{}t}[A]=2k_{\rm
I}[L][D]-...\,.$ (19)

Multiple extensions of Sanders’ model have been produced[71, 72, 73, 74] and
there are also other variance that cannot are not based on nucleotides, but on
peptides.[75, 76, 77, 78] If one regards these first polymerization reactions
as the first steps toward life, one could then say that homochirality emerges
as a consequence of life, and not as a prerequisite.[79, 80]

Figure 5: Gradual shrinking of isolated islands on one handedness (white),
leading to the eventual dominance of the other (gray).

###  5.2 Analogous mechanisms in MHD

The lessons learnt from astrobiology may well be applicable to other fields of
physics, and in particular to MHD. Examples were found in the context of the
magnetobuoyancy instability[81] and the Tayler instability.[82, 83] In those
cases, there are two unstable eigenfunctions that are helical and have
positive and negative helicities, respectively, but their growth rates are
equal. This process corresponds to autocatalysis. The nonlinearity in the MHD
equations, associated with the Lorentz force, corresponds to mutual
antagonism. The resulting amplitude equations in MHD are the same as in the
production of homochirality.[83] Another more recent example of this type has
been found in studies of the CME when the chiral chemical potential is
fluctuating around zero.[84, 85] Again, one chirality becomes eventually
dominant, and this choice depends on details in the initial conditions.

###  5.3 Spatially distinct domains of chirality

In spatially extended domains, the evolution equations for $[L]$ and $[D]$
attain additional spatial diffusion terms and become then similar to the
Fisher equation[86] which describes front propagation. In the present case,
once fronts of l and d polymers come into contact, the front between them
comes to a halt and cannot propagate any further, unless the front is curved.
If it is curved, the front continues to propagate in the direction of largest
curvature.[87] This leads in the end to small near-circular islands that
shrink until they disappear. This means that the enantiomeric excess changes
in a piecewise linear fashion.

Given that closed patches of one handedness always shrink and eventually
disappear, it is clear that the dominant chirality must in the end be that of
the outside of the last surviving patch. Thus, it is not necessarily the one
that was initially the most dominant one.

Figure 6: The number of infected $I(x,y)$ in a two-dimensional Cartesian plane
$(x,y)$, as obtained from model calculations,[88] showing (a) a circular
spreading center in the upper right corner, (b) the subsequent emergence of
eight additional spreading centers that (c) continue to grow, but with
decreasing $I(x,y)$ in their centers due to recovery or death. Note that the
total length of the periphery increases when the number of spreading centers
increases. Panels (d)–(e) show a case where spreading centers merge, so the
growth in the length of the periphery declines.  Figure 7: Square root of the
number $N$ of deaths, which is regarded as a proxy of the number of infected
that is more reliable than the reported number of SARS-CoV-2. Note the
piecewise linear growth in $N^{1/2}$, corresponding to a piecewise quadratic
growth.

A piecewise linear evolution is common to many spatially extended systems,
including those describing the spread of SARS-CoV-2 over the past two years.
Here, however, it is not the number $N$ itself, but its square root,
$N^{1/2}$. In Figure 6, we show the spatial geometry of a hypothetic spreading
center in the upper right corner. The speed of growth depends just on the
length of the periphery. At a later time, there will be new spreading centers,
so the total length of the periphery increases. This happened in the middle of
March 2020; see Figure 7. During the first part of the epidemic (denoted by a
red A), the evolution was comparatively slow and the disease was essentially
confined just to China. During the subsequent phase (denoted by a blue B) it
spread all over the world. This led to an increase in the total length of the
periphery. Later, different spreading centers began to emerge, so the total
periphery has now decreased again, and the growth has slowed down (denoted by
a green C), but it always remained piecewise quadratic and was never
exponential.[88] The spreading of SARS-CoV-2 is obviously no longer directly
related to the topic of chirality in astrophysics, but it is interesting to
see that the mathematics of front propagation in spatially extended domains is
similar to that of left and right handed life forms invading the early
Earth.[87]

##  6 Conclusions

In this work, we have sketched three rather different ways of achieving
chirality in astrophysics and astrobiology: externally driven, intrinsically
driven, and spontaneous chirality production. The first two mechanisms are
particularly relevant to fluid dynamics and magnetic fields, while the last
one of spontaneous chirality production is mainly relevant to astrobiology and
to the origin of life, but it remains hypothetic until one is able to find an
example of another genesis of life independent of that on Earth, for example
on Mars or on some of the icy moons in our solar system.[65]

The idea of propagating fronts of life forms of opposite handedness is
intriguing and it would be useful to reproduce this in the lab. This may not
be easy because such fronts propagate relatively rapidly under laboratory
conditions. In fact, their propagation resembles the propagation of
epidemiological fronts, such as the black death[89] and perhaps even SARS-
CoV-2.[88] However, the application to the early Earth implies much larger
spatial scales. Earlier work quoted half a billion years as a relevant time
scale.[87] However, if one thinks of the deep biosphere of Mars, it may not be
impossible to explain a possible detection of opposite chiralities of DNA, if
such should ever be observed on Mars or in its permafrost.

Alternatively, it is possible that the chirality of biomolecules is determined
through an external influence.[90] Such an influence would then, similarly to
the intrinsically driven chirality discussed in Sect. 4, be related to the
parity-breaking weak force. This possibility cannot easily be dismissed. In
particular, a 2% enantiomeric excess in favor of the l form has been found for
amino acids in the Murchison meteorite.[91] This was a pristine meteorite rich
in organics, as already evidenced by the smell reported by initial
eyewitnesses. On the other hand, those molecules are also susceptible to
contamination, while those not susceptible to contamination did not show any
enantiomeric excess.

Determining a global chirality that is the same throughout the entire Universe
would be a major discovery. Measuring the chirality through circular
polarization of GWs would likely be the most definitive proof of parity
invariance in the Universe.

## Acknowledgments

This work was supported in part through the Swedish Research Council, grant
2019-04234. I acknowledge the allocation of computing resources provided by
the Swedish National Allocations Committee at the Center for Parallel
Computers at the Royal Institute of Technology in Stockholm.

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# On Manifold Learning in Plato’s Cave:
Remarks on Manifold Learning and Physical Phenomena

Roy R. Lederman and Bogdan Toader
Yale University

###### Abstract

Many techniques in machine learning attempt explicitly or implicitly to infer
a low-dimensional manifold structure of an underlying physical phenomenon from
measurements without an explicit model of the phenomenon or the measurement
apparatus. This paper presents a cautionary tale regarding the discrepancy
between the geometry of measurements and the geometry of the underlying
phenomenon in a benign setting. The deformation in the metric illustrated in
this paper is mathematically straightforward and unavoidable in the general
case, and it is only one of several similar effects. While this is not always
problematic, we provide an example of an arguably standard and harmless data
processing procedure where this effect leads to an incorrect answer to a
seemingly simple question. Although we focus on manifold learning, these
issues apply broadly to dimensionality reduction and unsupervised learning.

##  I Background

The abundance of data in many applications in recent years allows scientists
to sidestep the need for parametric models and discover the structure of
underlying phenomena directly from some form of intrinsic geometry in the
measurements. Such concepts frequently appear in unsupervised learning,
manifold learning, non-parametric statistics and, more broadly, machine
learning. Often, a scientist may have in mind a concept of the “natural”
geometry or parametrization of the phenomenon; in other cases, they may
implicitly assume that only one such objective geometry exists even if they do
not know what it is. This paper aims to illustrate the difference between the
structure of observed data and some notion of natural or unique objective
structure. To this end, we offer a concrete example with an obvious underlying
natural geometry (up to symmetries) and demonstrate the existence of
discrepancies between the data and the natural variables, even in this benign
setting.

In our example, described more formally below, a simplified instance of a
physical phenomenon is represented by a rigid 3D model of a horse on a
spinning table. The measurement device is a fixed camera that takes images of
the object. The orientation angles of the horse are distributed uniformly.
Here, a natural variable is the angle at which the figure is oriented at the
time of the measurement. A simple example of a scientific question is to find
the mode of the distribution, which is intuitively the most prevalent
orientation angle (we know that the correct answer is that the distribution is
uniform and, therefore, we do not expect to find a clear mode). Since this is
meant to be a simplified, intuitive version of a generic problem with no
obvious underlying model, we consider generic algorithms and forgo in advance
image analysis and computer vision methods that use of the special properties
of images and the specific rotating motion of the object.

This benign task yields results that we find surprising yet predictable. The
naive analysis discovers clear modes of the distribution, which are
inconsistent with the true uniform distribution. In Appendix -A3, we
demonstrate that these modes are not invariant to the measurement modality.

In our discussion, we explain the reasons for the experiment’s results and
refer the reader to existing work on special cases where the problem can be
corrected. However, there is no method for correcting the problem in the
general case. We conclude by pointing to where care should be taken in
defining the problem and using the output in downstream tasks.

We emphasize that this paper aims to highlight an omission that we observe in
the practical use of manifold-related machine learning algorithms in
applications. The purpose of this paper is not to advocate against these
methods but rather to suggest that care should be taken in stating and
interpreting their output.

##  II The problem

The mathematical setting of the experiment is simple: let
$\operatorname{\mathcal{X}}\subset\operatorname{\mathbb{R}}^{d}$ and
$\operatorname{\mathcal{Y}}\subset\operatorname{\mathbb{R}}^{D}$ be two
manifolds with $d\ll D$ and
$f:\operatorname{\mathcal{X}}\rightarrow\operatorname{\mathcal{Y}}$ be a
diffeomorphism. We refer to $\operatorname{\mathcal{X}}$ and
$\operatorname{\mathcal{Y}}$ as the phenomenon manifold and the measurement
manifold respectively and to $f$ as the measurement function. In our simple
experiment, the phenomenon manifold $\operatorname{\mathcal{X}}$ is the one-
dimensional torus representing the orientation of the horse with respect to a
fixed frame of reference (independent of the camera), the measurement function
$f$ outputs an image of the horse as captured by the camera, and the
measurement manifold $\operatorname{\mathcal{Y}}$ is the manifold of images
obtained by the camera. In particular, a sample
$x\in\operatorname{\mathcal{X}}$ is the angle of the horse at a specific point
in time, and the corresponding measurement $f(x)\in\operatorname{\mathcal{Y}}$
is the image of the horse at the same point in time. In a typical setting, we
are given a large set of measurements
$\left\\{y_{i}\right\\}_{i=1}^{n}\subset\operatorname{\mathcal{Y}}$ of a set
of samples $\\{x_{i}\\}_{i=1}^{n}$ drawn from a distribution
$\operatorname{\mathcal{D}}$ on $\operatorname{\mathcal{X}}$. Here, we take
the distribution $\operatorname{\mathcal{D}}$ of the orientation angles of the
horse to be uniform, which would be unknown in an actual experiment. We only
have access to the measurements $\\{y_{i}\\}_{i=1}^{n}$ (the images of the
horse), which we assume to be noise-free for simplicity, and we are interested
in uncovering the low-dimensional organization of the samples of angles
$\\{x_{i}\\}_{i=1}^{n}$, for example, their empirical distribution on
$\operatorname{\mathcal{X}}$. The setting of this numerical experiment is
illustrated in Figure 1. For simplicity and concreteness, we apply common
techniques to answer a simple question: what is the most dominant physical
state? We know that the ground truth answer is, in this case, that there is no
dominant state; the data are generated with uniform distribution over the
orientation angles of the horse.

Figure 1: The phenomenon manifold $\operatorname{\mathcal{X}}$ is a one-
dimensional torus corresponding to the in-plane orientation angle of a rigid
object rotating around the $z$-axis, and the measurement manifold
$\operatorname{\mathcal{Y}}$ is the manifold of images of the object as
captured by a camera at a fixed location.

We follow a common practice of assuming a low-dimensional structure and apply
a manifold learning algorithm. This produces the map
$\rho:\operatorname{\mathcal{Y}}\to\operatorname{\mathcal{Z}}$, which yields a
low-dimensional embedding of the measurements
$\rho(y_{i})\in\operatorname{\mathcal{Z}}$ for $i=1,\ldots,n$ and
$\operatorname{\mathcal{Z}}\subset\operatorname{\mathbb{R}}^{s}$ with $d\leq
s\ll D$. In our experimental setting, the low-dimensional assumption is
clearly true: the orientation angles of the horse lie on the one-dimensional
torus manifold, while the measurements are clearly high-dimensional (the
number of pixels of each image). For simplicity, in our numerical experiment,
we use the diffusion maps algorithm, whose theoretical properties are well
understood [1, 2], and we retain only the first two diffusion coordinates, a
standard practice in this simple case. The output we expect to see is an
embedding of the one-dimensional torus in $\mathbb{R}^{2}$: a circle.

It is common in applications to apply a machine learning or manifold learning
algorithm to the measurements $\\{y_{i}\\}_{i=1}^{n}$, and consider the low-
dimensional embeddings $\\{\rho(y_{i})\\}_{i=1}^{n}$ to be a proxy for the
geometry of the actual samples $\\{x_{i}\\}_{i=1}^{n}$; the potential effects
of the measurement function $f$ are omitted. The aim of this manuscript is to
demonstrate that even in the most benign setting, the measurements distort the
physical problem in a way that can impact a seemingly straightforward
analysis.

Many algorithms for manifold learning and visualization have been developed
over the years and have been found useful in applications. Often, these
algorithms start with the pair-wise distances $\|y_{i}-y_{j}\|$ (in some
norm), for $i,j=1,\ldots,n$, as a measure of (inverse) similarity, but diverge
in their precise formulation of the problem. One of the notable departures
from this approach is the use of the latent space estimated in the training of
deep neural networks as the manifold embedding, with the variational
autoencoder (VAE) [3] being one of a number of popular approaches.

The diffusion maps algorithm produces coordinates that are related to the
geometry of the data through a diffusion operator on the data manifold. While
there are technical nuances in the metric defined by diffusion maps (and other
algorithms) and in retaining only two dimensions, this example is particularly
benign, symmetric, and without boundary effects. Therefore, one expects the
leading eigenvectors of the discretized diffusion operator to preserve the
local geometry of the data (up to scaling). For a formal description of the
diffusion maps algorithm and its properties, see [1, 2]. One of the appealing
properties of the diffusion maps algorithm is that it is (asymptotically)
invariant to the local density of the data and captures only its local
geometry. This property and the algorithm’s explicit relationship to the
geometry of the data made it a good choice for our experiments.

Indeed, a diffusion map of the points on $\operatorname{\mathcal{X}}$
preserves the geometry and the uniform distribution (shown in Appendix -A1).
However, our measurement function is not necessarily an isometry (even up to
scaling), and therefore, it distorts the geometry and the local pair-wise
distances.

The low-dimensional embedding obtained by applying the diffusion maps
algorithm to a dataset $\\{y_{i}\\}_{i=1}^{n}$ of size $n=1000$ and ambient
dimension $D=108000$ ($180\times 200$ size images with $3$ color channels) in
our experimental setting111 The code and dataset to reproduce the numerical
experiments described in this paper can be downloaded from
https://github.com/bogdantoader/ManifoldLearningInPlatosCave.  is shown in
Figure 2. Both panels show a scatter plot using the first two embedding
coordinates given by the diffusion maps algorithm. The points in panel (a) are
colored according to the true angle $x_{i}$ for $i=1,\ldots,n$. Visually, it
appears that the algorithm reveals the correct topology and it organizes the
images correctly by their angle. It is compelling to say that the embedding is
a good approximation of the angles (up to shift). However, taking a closer
look at the distribution of the points in panel (b), we see that that their
local density222 We define the local density at a point $z_{i}$ on the
embedded manifold $\operatorname{\mathcal{Z}}$ as the number of points $z_{j}$
in the ball centered at $z_{i}$ with radius $r$, for a given $r>0$, normalized
so that the densities sum up to one, and using the metric on
$\operatorname{\mathcal{Z}}$. Since in these experiments, we used the
diffusion maps algorithm with a two-dimensional latent space, the diffusion
metric on $\operatorname{\mathcal{Z}}$ corresponds to the Euclidean metric on
$\mathbb{R}^{2}$.  has not been preserved on the embedding manifold
$\operatorname{\mathcal{Z}}$: while the distribution
$\operatorname{\mathcal{D}}$ of the points $\\{x_{i}\\}_{i=1}^{n}$ on
$\operatorname{\mathcal{X}}$ is uniform (by construction!), the distribution
of the embedded points $\\{\rho(y_{i})\\}_{i=1}^{n}$ on
$\operatorname{\mathcal{Z}}$ is not uniform. Moreover, the distribution of the
embedded points has two clear modes, with no indication that they are an
artifact of the analysis.

An additional experiment showing how the distribution of the embedded points
varies when the viewing angle is changed is described in Appendix -A3, and the
specific implementation of the diffusion maps algorithm that we used in our
experiments is presented in Appendix -B.

(a) True angle

(b) Local density

Figure 2: Low-dimensional embedding of the images of the spinning horse. The
coloring is given by the true orientation angle of the horse in panel (a) and
the local density of points ($r=0.05$) in panel (b).

##  III Discussion

In the previous section, we empirically showed how the distribution of the
points on the embedding manifold $\operatorname{\mathcal{Z}}$ does not reflect
the true distribution of the points on the phenomenon manifold
$\operatorname{\mathcal{X}}$: the distribution on $\operatorname{\mathcal{Z}}$
has two distinct modes, while the distribution on $\operatorname{\mathcal{X}}$
is uniform. To see that this is a metric-related issue, it is worth examining
the modes of the distribution on $\operatorname{\mathcal{Z}}$.

In Figure 3, we show measurements at a high and a low-density point on
$\operatorname{\mathcal{Z}}$. It is revealed that the high-density regions
correspond to images where the three-dimensional object is perpendicular (or
nearly perpendicular) to the viewing direction of the camera, while the low-
density regions correspond to the object facing toward or away from the
camera. This is because, according to our chosen metric on
$\operatorname{\mathcal{Y}}$ (i.e., the Euclidean norm on the space of
vectorized images), a small difference $\Delta x$ between two angles in
$\operatorname{\mathcal{X}}$ is not transformed to the same distance in
different regions of $\operatorname{\mathcal{X}}$: two images of the object
facing the camera that differ by $\Delta x$ have a larger Euclidean distance
than two images of the object facing sideways that are separated by the same
angle. The metric based on the measurements alone does not account for the
distortion introduced by the measurement function $f$ on the true metric on
$\operatorname{\mathcal{X}}$, namely the wrap-around distance on $[0,2\pi)$.

Figure 3: The low-dimensional embedding with example images corresponding to
samples from the estimated distribution. The image to the left of the
embedding plot is chosen to be at a low local density in the embedding, and
the image to the right is chosen to be at the maximum density point.

The discrepancy between the metric on the phenomenon manifold, which is the
metric we want to recover, and the arguably arbitrary metric produced by the
measurement modality can be corrected in some special cases. For example, when
bursts of measurements around each point on $\operatorname{\mathcal{X}}$ are
available, one can use the Jacobian of the measurement function to define
metrics that are invariant to the measurement modality (see, for example, [4,
5, 6, 7, 8]). Such metrics might still not be the “desired” metrics we want to
conceptualize, but they are “Platonic” in the sense that they are defined on
the phenomenon manifold $\operatorname{\mathcal{X}}$ and are invariant to the
arbitrary measurement function. Other works such as [9, 10, 11, 12, 12]
correct the metric distortion introduced by the embedding $\rho$ from the
measurement manifold $\operatorname{\mathcal{Y}}$ to the embedded manifold
$\operatorname{\mathcal{Z}}$; these works do not correct the discrepancy
between the measurements and the metric on the phenomenon manifold.

We emphasize that the problem illustrated here is not due to a failure of the
diffusion maps or other algorithms; the algorithm performs as expected and
characterizes the measurement manifold very well. However, the metric of this
measured manifold is incompatible with the natural metric of in-plane rotation
angles. As a result, we identify modes of the distribution in the measurement
space, but these do not correspond to modes of the underlying distribution of
angles.

We note that the problem discussed here is not unique to the diffusion maps
algorithm or the setup we chose; in fact, other algorithms are not as well-
understood as diffusion maps, and applications are rarely as simple as our
illustrative example. Many modern algorithms add layers of complexity to the
problem. For instance, deep learning approaches that generate latent
variables, such as VAEs, are often combined with more standard manifold
learning algorithms to obtain low-dimensional data representations. In [13,
14], the distortions introduced by popular algorithms like t-SNE and UMAP are
analyzed in the context of single-cell genomics, although the focus is on the
discrepancy between the high dimension of the measurement space and the very
low dimension (2 or 3) of the embedding space, rather than on the choice of
metric. While such algorithms provide valuable new insights into datasets,
practitioners should be aware that the results they generate, even when they
perform as intended, may have a subtle relation to the “Platonic” physical
reality. These outputs should arguably mainly be used for visualization and
confirmed by other means. Indeed some of the original work on popular non-
linear dimensionality reduction algorithms defines them as tools for
visualization [15, 16].

##  IV Conclusions

This paper illustrates one of the discrepancies between the measured manifold
and a perceived natural parametrization of the underlying phenomenon. In
addition, Appendix -A3 demonstrates how this discrepancy depends on the
measurement modality and how the measured manifold is not invariant to
measurements. The discrepancy presented here is by no means the only type of
discrepancy; we defer the discussion of additional effects to future work.
While the existence of this discrepancy is a natural consequence of various
mathematical formulations of manifold learning problems (with the exception of
special cases where the metric can be corrected), it is occasionally omitted,
which may lead to incorrect and inconsistent answers to seemingly simple
scientific questions. In the absence of a general solution to the problem, we
suggest the following points to consider when using these methods.

  * •

A good rule of thumb is that manifold learning and dimensionality reduction
can provide (when they “work”) an embedding, but they may not provide the
embedding (that we might have in mind). In fact, without a good definition of
the desired embedding, the embedding is not unique.

  * •

Sometimes, the effects can be controlled if there is knowledge of the
structure of the measurement function (e.g., Lipschitz constant). However,
nuances in definitions of the output of algorithms, the increased complexity
of algorithms, and the practice of layering algorithms on top of each other
may make it much more difficult to control such effects. In some special
cases, additional measurements may allow one to reverse the effect [4, 5, 6,
7, 8].

  * •

In many (but not all) applications, the inferred manifold may reveal enough
about the topology of the problem, or the distortion in the metric might be
sufficiently small to be a sufficiently good proxy for the geometry. What is
“sufficiently good” may depend on the downstream task. For example, the low-
dimensional manifold may be a starting point for an analysis by an expert,
regression or careful clustering, suspected outliers detection, and even for
identification of clear modes. It may not be as helpful for aligning data
collected using different modalities (or even different algorithms applied to
the same data) with different distortions (see Appendix -A3), or for certain
analyses of free energy associated with the distribution.

##  V Acknowledgments

This work was supported by the grants NIH/R01GM136780 and
AFOSR/FA9550-21-1-0317, and by the Simons Foundation.

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###  -A Additional experiments

This appendix shows several plots and numerical experiments that further
illustrate the discrepancy between the phenomenon manifold and the measurement
manifold.

####  -A1 Manifold learning on the phenomenon manifold

We have shown in the main text of the paper how applying a manifold learning
algorithm to the points on $\operatorname{\mathcal{Y}}$ leads to modes in the
data, while we know that the samples are uniformly distributed on the
phenomenon manifold $\operatorname{\mathcal{X}}$ (the orientation angles of
the horse). In this section, we present the same manifold learning algorithm
applied directly to the points in $\operatorname{\mathcal{X}}$ (which are not
available to us as measurements in our original problem setup) as a benchmark
for comparison to the results in the main text, where we apply the algorithm
to the measurements. In Figure 4, we ran the diffusion maps algorithm to the
set of angles that generated the images, where we represented each angle by a
complex number with the magnitude of one and the given angle, and we used the
Euclidean distance between them as the distance on
$\operatorname{\mathcal{X}}$. As expected, the embedded points have constant
local density (i.e., they are uniformly distributed).

(a) True angle

(b) Local density

Figure 4: The embedding resulting from applying diffusion maps to the angles
directly, represented as complex numbers of magnitude one. The coloring is
given by the true angle in panel (a) and the local density with $r=0.05$ in
panel (b).

####  -A2 Local density of the measured data points

The central claim of this paper is that applying manifold learning to measured
data without careful attention to the metric on $\operatorname{\mathcal{Y}}$
can lead to wrong conclusions about certain questions. The point where metric
distortions due to the measurement function are introduced in the processing
pipeline is when computing pair-wise distances between the points on
$\operatorname{\mathcal{Y}}$. As a further check that this is indeed the case,
in Figure 5 we show the local density of the images themselves (the points on
$\operatorname{\mathcal{Y}}$): the coordinates of each image are its embedding
coordinates (the same as in the main text), and the coloring is given by the
local density of the images in panel (a) and the local density of the
embeddings in panel (b). The radius $r$ of the ball used to approximate the
local density in (a) was chosen so that the maximum value of the local density
is approximately the same as the maximum value of the local density in (b).
Figure 5 shows that the same modes seen in the embedding are also present in
the measured data. The specific value may not be numerically identical due to
subtleties in the definition of the embedding and the nature of the
approximation. However, the same kind of effect is clearly visible.

(a) Local density on $\operatorname{\mathcal{Y}}$

(b) Local density on $\operatorname{\mathcal{Z}}$

Figure 5: Local density of the points on $\operatorname{\mathcal{Y}}$ in (a)
and on $\operatorname{\mathcal{Z}}$ in (b). The coordinates of the points are
given by the embedding in $\operatorname{\mathcal{Z}}$ in both plots. The
radius for computing the local density on $\operatorname{\mathcal{Y}}$, $r=9$,
was chosen so that the maximum value of the density matches the maximum value
of the density on $\operatorname{\mathcal{Z}}$.

####  -A3 Two modalities and non-uniqueness

To further illustrate the problem raised in this article, we now consider a
slightly modified setup to demonstrate the non-uniqueness of the
parametrization. Instead of one camera, we take photographs of the spinning
rigid object using two different cameras placed at two distinct locations. We
denote the two measurement functions by $f_{1}$ and $f_{2}$ and the two
datasets by $\\{y_{i}^{1}\\}_{i=1}^{n}\subset\operatorname{\mathcal{Y}}_{1}$
and $\\{y_{i}^{2}\\}_{i=1}^{n}\subset\operatorname{\mathcal{Y}}_{2}$. The
images $y_{i}^{1}$ and $y_{i}^{2}$ at index $i$ taken by the two cameras
respectively correspond to the same ground truth point $x_{i}$. This setup is
illustrated in Figure 6. The ambient dimension is $D=108000$, the number of
images taken by each camera is $n=1000$, and the measurements are taken at
equal time intervals, so the distribution of the data on the phenomenon
manifold $\operatorname{\mathcal{X}}$ is uniform.

Figure 6: Setup of the numerical experiment with two measured datasets. Using
two cameras at different locations, we collect two distinct sets of images on
the measurement manifolds $\operatorname{\mathcal{Y}}_{1}$ and
$\operatorname{\mathcal{Y}}_{2}$, respectively.

We apply the diffusion maps algorithm separately to each dataset,
corresponding to samples from the two measurement manifolds
$\operatorname{\mathcal{Y}}_{1}$ and $\operatorname{\mathcal{Y}}_{2}$. The
resulting two-dimensional embeddings are shown in Figure 7: the left column
shows the embedding $\\{\rho_{1}(y_{i}^{1})\\}_{i=1}^{n}$ obtained from the
measurements $\\{y_{i}^{1}\\}_{i=1}^{n}\subset\operatorname{\mathcal{Y}}_{1}$
and the right column shows the embedding $\\{\rho_{2}(y_{i}^{2})\\}_{i=1}^{n}$
obtained from
$\\{y_{i}^{2}\\}_{i=1}^{n}\subset\operatorname{\mathcal{Y}}_{2}$. In each
panel, we show a scatter plot using the first two embedding coordinates given
by the diffusion maps algorithm. We denote by $\operatorname{\mathcal{Z}}_{1}$
and $\operatorname{\mathcal{Z}}_{2}$ the resulting low-dimensional manifolds.
Similarly to the one-camera experiment described in the main text, the one-
dimensional torus topology of the orientation angles of the object is
correctly identified (panels (a) and (b)). However, panels (c) and (d) show
that the metric is distorted. In particular, the distributions of the images
in both datasets are incorrectly shown to have two modes (we know that the
true distribution is uniform). Moreover, the modes are not compatible: the
mode observed in one camera does not correspond to the same true angles as
those observed in the other camera. This can be seen in Figure 8, where images
from high and low-density regions on $\operatorname{\mathcal{Z}}_{1}$
correspond to seemingly arbitrary points on $\operatorname{\mathcal{Z}}_{2}$,
whose high and low-density regions correspond to different orientations on
$\operatorname{\mathcal{X}}$. This is, of course, not surprising, given that
each camera takes the photographs from different directions and considering
the symmetries in this problem.

In this paper, we showed how attempts to solve a simple problem, such as
identifying the dominant state of a system, can lead to incorrect answers when
applying manifold learning to the measured data. While the experiment
presented in the main text of the article shows that the answer we obtain can
be incorrect in a non-obvious way, the two-camera experiment presented in this
appendix demonstrates that different measurement modalities can produce
different answers, with no way to compare the quality of the two answers
objectively. More generally, this shows that the estimated measurement
manifold is not unique and not invariant to the measurement modality.

This experiment corresponds to several relatively common real-life settings.
One example is when multiple different algorithms are applied to the same
data, perhaps with different processing pipelines. The processing pipelines
are analogous to our different cameras and may not produce the same result.

Another real-life setting is when attempting to align measurements of an
underlying phenomenon, acquired using two distinct modalities, calibrated
differently and possibly taken on different days. This is the case, for
example, when the data is collected in two separate batches, potentially in
different laboratories. In this case, even if the two datasets are assumed to
have the same distribution, batch effects are present due to potentially
different experimental settings. Our experiments show that a rigid
transformation cannot align embeddings obtained from such datasets.

(a) Left camera, true angle

(b) Right camera, true angle

(c) Left camera, local density

(d) Right camera, local density

Figure 7: Embeddings obtained using images from the left camera (left column)
and the right camera (right column). The coloring is given by the true angle
in the top row and the local density of points ($r=0.05$) in the bottom row.
Figure 8: The embeddings with example images from the estimated distributions.
The images in the top row correspond to a point in a high-density region of
the left camera embedding, and the images in the bottom row correspond to a
point in a low-density region of the left camera embedding. In each row of
images, we show the top view of the object displaying the true orientation
angle (left), the object as seen by the left camera (middle), and the object
as seen by the right camera (right).

####  -A4 Top view measurements

In the previous experiment, we showed that different measurement functions
lead to embeddings where the metric is distorted in different ways without
knowing which one is more accurate. To further strengthen this argument, we
show an example where the measurement function distorts the distances but
preserves the local geometry. The images of the rotating horse are captured
from the top, and the resulting embedding and density are shown in Figure 9.
In practice, there is no way of knowing, only from the data, that we are in a
case where the local geometry on $\operatorname{\mathcal{X}}$ is preserved.
This experiment and the one in Appendix -A3 reinforce that the embedding we
obtain depends heavily on how the measurements are taken and that the solution
we expect to see is not objectively better than other possible solutions,
given the data.

(a) True angle

(b) Local density

Figure 9: Low-dimensional embedding of the images of the spinning horse viewed
from the top. The coloring is given by the true orientation angle of the horse
in panel (a) and the local density of points ($r=0.05$) in panel (b).

###  -B Diffusion maps

For completeness, we present the diffusion maps algorithm used throughout the
article, as described in [1] and adapted in [17]. The phenomena discussed in
this paper are not specific to diffusion maps. Broadly interpreted, these
issues appear in many machine learning and manifold learning problems in one
form or another, except for special cases where they can be corrected or
“defined away.”

Input: Samples $\\{y_{i}\\}_{i=1}^{n}\subset\operatorname{\mathcal{Y}}$, a
metric $d(y_{i},y_{j})$ on $\operatorname{\mathcal{Y}}$, kernel width
$\sigma$.

Output: Low-dimensional embedding coordinates
$\\{z_{i}\\}_{i=1}^{n}\subset\mathbb{R}^{s}$ and diffusion distances
$\tilde{d}(z_{i},z_{j})$.

  1. 1.

Compute the similarity matrix $W$:

$W_{ij}=\exp\left(-\frac{d(y_{i},y_{j})^{2}}{\sigma}\right),\forall
i,j=1,\ldots,n.$

  2. 2.

Compute the diagonal normalization matrix

$Q_{ii}=(\sum_{j=1}^{n}W_{ij})^{-1}.$

  3. 3.

Normalize the kernel $\tilde{K}=QWQ$.

  4. 4.

Compute the second diagonal normalization matrix

$\tilde{Q}_{ii}=(\sum_{j=1}^{n}\tilde{K}_{ij})^{-1/2}.$

  5. 5.

Normalize the kernel $K=\tilde{Q}\tilde{K}\tilde{Q}$.

  6. 6.

Compute the $n$ eigenvectors $u_{0},u_{1},\ldots,u_{n-1}$ and eigenvalues
$\lambda_{0},\lambda_{1},\ldots,\lambda_{n-1}$ of $K$.

  7. 7.

The $s$-dimensional embedding of each point $y_{i}$ is

$z_{i}=\left(\lambda_{1}\frac{u_{1}(i)}{u_{0}(i)},\lambda_{2}\frac{u_{2}(i)}{u_{0}(i)},\ldots,\lambda_{s}\frac{u_{s}(i)}{u_{0}(i)}\right)^{T}.$

  8. 8.

The diffusion distance is $\tilde{d}(z_{i},z_{j})=\|z_{i}-z_{j}\|_{2}.$