# Invariants in Polarimetric Interferometry: a non-Abelian Gauge Theory

Joseph Samuel<EMAIL_ADDRESS>International Centre for Theoretical Sciences,
Bengaluru 560089, Karnataka, India  Raman Research Institute, Bengaluru
560080, Karnataka, India  Rajaram Nityananda<EMAIL_ADDRESS>Azim Premji University, Bikkanahalli Main Road, Sarjapura, Bengaluru 562125,
Karnataka, India  Nithyanandan Thyagarajan<EMAIL_ADDRESS>https://tnithyanandan.wordpress.com/ CSIRO, Space & Astronomy, P. O. Box 1130,
Bentley, WA 6102, Australia  National Radio Astronomy Observatory, 1003
Lopezville Rd, Socorro, NM 87801, USA

###### Abstract

The discovery of magnetic fields close to the M87 black hole using Very Long
Baseline Interferometry (VLBI) by the Event Horizon Telescope collaboration
utilized the novel concept of “closure traces”, that are immune to element-
based aberrations. We take a fundamentally new approach to this promising tool
of polarimetric VLBI, using ideas from the geometric phase and gauge theories.
The multiplicative distortion of polarized signals at the individual elements
are represented as gauge transformations by general $2\times 2$ complex
matrices, so the closure traces now appear as gauge-invariant quantities. We
apply this formalism to polarimetric interferometry and generalize it to any
number of interferometer elements. Our approach goes beyond existing studies
in the following respects: (1) we use triangular combinations of correlations
as basic building blocks of invariants, (2) we use well-known symmetry
properties of the Lorentz group to transparently identify a complete and
independent set of invariants, and (3) we do not need auto-correlations, which
are susceptible to large systematic biases, and therefore unreliable. This set
contains all the information, immune to corruption, available in the
interferometer measurements, thus providing important robust constraints for
interferometric studies.

Gauge theories; Gauge theory techniques; Geometric & topological phases;
Geometrical & wave optics; Group theory; Imaging & optical processing;
Interferometry; Lattice gauge theory; Lorentz symmetry; Mathematical physics;
Non-Abelian gauge theories; Polarization of light; Radio, microwave, & sub-mm
astronomy

††preprint: APS/123-QED

The measurement of coherence of fields is an important concept with
applications in many disciplines of physics. Radio astronomers measure
coherence by correlating radio signals received by an interferometer array and
characterize the morphology of radio emission received from the sky [1]. The
signals received at each telescope are usually corrupted due to propagation
effects and local imperfections in the array receiver elements. Finding
interferometric invariants, quantities that are immune to this corruption, and
so accurately reflect the true structure of the sky emission, is of
significant value. This is the principal focus of this paper.

The subject of closure phases and closure amplitudes, two popular
interferometric invariants in astronomy, has a long history [2, 3]. Concepts
analogous to closure invariants (interferometric invariants defined on closed
loops) occur in other areas like speckle interferomentry [4], crystallography
[5] and quantum mechanics [6, 7]. Our approach, as in the recent work [8], is
inspired by the geometric phase [7, 9] of quantum physics and classical optics
[10, 11]. A common thread that ties all these diverse problems together is
gauge theory. We adapt the general framework to the problem of closure
invariants in astronomy.

Closure invariants in co-polar correlations, in which signals of the same
polarization are correlated between all pairs of telescopes, are well studied
in both theory [12, 1] and practice [13, 14]. In a recent significant
extension to polarimetric measurements, “closure traces” were introduced and
explored in detail for a system of four array elements [15]. These closure
traces were used in studying the magnetic properties near the event horizon of
the supermassive black hole at the center of the galaxy M87 [16]. Generalizing
and extending this work [15] on polarimetric interferometry and closure
invariants, our paper applies the gauge theory framework to the group of
nonsingular $2\times 2$ matrices (known to mathematicians as
$\textrm{GL}(2,\mathbb{C})$) and Lorentz groups. We go beyond earlier work by
transparently determining the complete and independent set of invariants from
an $N$-element interferometer array. We show that auto-correlation
measurements, which are highly susceptible to systematic errors111In radio
astronomy, auto-correlations are problematic because of their susceptibility
to systematic effects caused by emission unrelated to the object being imaged
(e.g., atmospheric emission and radio frequency interference) and the gain
errors affecting these strong unwanted signals., are not necessary. If
available, they are easily included in the scheme. This should be useful in
future mapping of polarized sources, especially using long baseline
interferometry, where calibration is challenging.

Notation.—We use indices $a,b=0,\ldots,N-1$ to label the array elements, and
indices $p,q=1,2$ to label the two polarizations. $2\times 2$ and $4\times 4$
matrices are shown in uppercase boldface. 2-element vectors and 4-vectors are
written using lowercase boldface in regular and italicized fonts,
respectively. For 4-vectors and $4\times 4$ matrices written in component
form, we use relativity conventions, like the Einstein summation convention
for repeated Greek indices, which range over $0,1,2,3$.

Polarimetric Interferometry.—Consider an interferometer array with $N$
elements. The $p$-th component of polarization (in some orthonormal basis) of
the electric field incident on element, $a$, is denoted by $e_{a}^{p}$,
represented in amplitude and phase by a complex 2-element column vector,
$\mathbf{e}_{a}$. The true correlation matrix between the array elements is
obtained via pairwise cross-multiplication and averaging,
$\mathbf{S}_{ab}\coloneqq\left\langle\mathbf{e}_{a}\mathbf{e}_{b}^{\dagger}\right\rangle$,
where, $\dagger$ denotes a conjugate-transpose operation, and
$\langle\cdot\rangle$ denotes the average.

Due to the propagation medium and the non-ideal measurement process, the
incoming amplitudes are corrupted by an element-based linear transformation –
a general $2\times 2$ complex matrix, $\mathbf{G}_{a}$. The off-diagonal
entries of $\mathbf{G}_{a}$ represent leakage between the two polarized
receiver channels at array element $a$. The measured amplitudes,
$\mathbf{v}_{a}$, are related to the true amplitudes, $\mathbf{e}_{a}$, by
$\mathbf{v}_{a}=\mathbf{G}_{a}\mathbf{e}_{a}$.

The corrupted correlation matrix for a pair of elements, $(a,b)$, is
constructed from the measurements as

$\displaystyle\mathbf{C}_{ab}$
$\displaystyle=\left\langle\mathbf{v}_{a}\mathbf{v}_{b}^{\dagger}\right\rangle=\mathbf{G}_{a}\left\langle\mathbf{e}_{a}\mathbf{e}_{b}^{\dagger}\right\rangle\mathbf{G}_{b}^{\dagger}=\mathbf{G}_{a}\mathbf{S}_{ab}\mathbf{G}_{b}^{\dagger}\,.$
(1)

$\mathbf{G}_{a}$, $\mathbf{C}_{ab}$, and $\mathbf{S}_{ab}$ are $2\times 2$
matrices. Generically, their eigenvalues are non-zero and we will assume this
to be true. These matrices are then invertible and so belong to the general
linear group, $\textrm{GL}(2,\mathbb{C})$. Evidently,
$\mathbf{C}_{ab}^{\dagger}=\mathbf{C}_{ba}$. The auto-correlations,
$\mathbf{A}_{aa}\coloneqq\mathbf{C}_{aa}$, are Hermitian and positive definite
(strictly positive eigenvalues). We use a special symbol to distinguish them
from general cross-correlations, $\mathbf{C}_{ab}$.

Our objective is to construct quantities which are immune to the corruptions
and actually reflect the true coherence properties of the source of radio
emission, rather than local conditions (represented by $\mathbf{G}_{a}$) at
the interferometer elements. This problem is mathematically related to the
“gauge” theories of fundamental interactions, such as electromagnetism (EM) or
the nuclear force. We regard multiplication of the true signal by the local
gains, $\mathbf{G}_{a}$, as gauge transformations. We wish to eliminate these
spurious effects introduced by $\mathbf{G}_{a}$ and identify a maximal,
independent set of gauge-invariant quantities (independent of
$\mathbf{G}_{a}$), which we will call “closure invariants”. The gains are
elements of $\textrm{GL}(2,\mathbb{C})$, which is not unitary, resulting in
features not seen in the usual gauge theories of particle physics. The case of
co-polar observations is described in detail in [1, 12], where the gains are
just nonzero complex numbers, and the gauge group is
$\textrm{GL}(1,\mathbb{C})$, which is Abelian — a commutative group. In a
companion paper [Paper I; 17], we have introduced these ideas in the simpler
context of $\textrm{GL}(1,\mathbb{C})$, which serves as a stepping stone to
the polarized theory. The current paper deals with full polarimetric
measurements governed by the gauge group $\textrm{GL}(2,\mathbb{C})$. An
important difference between the two cases is that matrices
$\textrm{GL}(2,\mathbb{C})$ group do not, in general, commute. This introduces
a level of complexity not seen in the co-polar (Abelian) case [17].

Counting Invariants.—How many independent closure invariants can we expect to
find? The number of true cross-correlations, $\mathbf{S}_{ab}$, in the
measured cross-correlations, $\mathbf{C}_{ab}$, is equal to the number of
array element pairs, $N(N-1)/2$, which are assumed to be non-redundant in this
paper. Each $\mathbf{C}_{ab}$ gives us one complex $2\times 2$ matrix, with
$8$ real parameters. If auto-correlations are also measured, we would have to
add $4n_{\textrm{A}}$ to this count, since $\mathbf{A}_{aa}$ is Hermitian and
has $4$ real parameters, and $n_{\textrm{A}}$ is the number of true auto-
correlations in the measurements. When the object occupies a small fraction of
the field of view, all elements would measure essentially the same auto-
correlation (with an inherent assumption of spatial stationarity). Therefore,
$\mathbf{S}_{aa}=\mathbf{S}_{bb}$ for all $a,b$, and we can set
$n_{\textrm{A}}$ to zero or one without loss of generality. Thus, if
$\mathcal{M}$ is the space of all measured correlations, its dimension is the
total number of real parameters in the measured correlations,
$\mathrm{dim}(\mathcal{M})=8N(N-1)/2+4\,n_{\mathrm{A}}$.

Each of the $N$ gain matrices, $\mathbf{G}_{a}$, is described by 4 complex
numbers (or, 8 real numbers), resulting in $8N$ real parameters in total. The
gains $\mathbf{G}_{a}$ from the $N$ array elements corrupt the signals
according to Eq. (1) and the $\mathbf{C}_{ab}$ could lie anywhere on a surface
of points containing $\mathbf{S}_{ab}$ (see Fig. 1). Any function of the
measurements which is constant on each surface is immune to corruption by the
gains. These are the interferometric invariants we seek. The number of
independent interferometric invariants is the dimension of $\mathcal{M}$ minus
the dimension of the surface.

Figure 1: A schematic representation of the space of correlations using a dark
three-dimensional region. If the true correlation due to the source structure
is at the point marked $S_{ab}$, the measured correlation, $C_{ab}$, could lie
anywhere on the gray two dimensional surface because of gain distortions. For
$N>3,n_{\textrm{A}}=0$, the correlation space and the surface have real
dimensions $4N(N-1)$ and $8N-1$, respectively. The figure shows these spaces
as three- and two-dimensional, respectively, for illustrative purposes.

The dimension of the surface can at most be the dimension of the gains
$\mathbf{G}_{a}$, which is $8N$. It could be smaller if there are gain
variations which do not corrupt the measured correlations222In gauge theory,
these correspond to a subgroup (called the stabilizer), which acts trivially
(i.e., does not affect the measured correlations).. We refer to these as Non-
Corrupting Gains, NCGs for short. As a result, we would expect

$\displaystyle N_{\mathcal{I}}$ $\displaystyle=4N^{2}-12N+4n_{\textrm{A}}+s$
(2)

independent invariants, where $s$ is the number of parameters in NCGs
(dimension of the stabilizer subgroup). One well-known example of an NCG is a
constant phase shift applied to all the signals on all the interferometer
elements, which would not affect $\mathbf{C}_{ab}$. Such a uniform phase shift
applied to all elements belongs to the $\textrm{U}(1)$ subgroup, which forms
the entire stabilizer group. This is the only NCG ($s=1$) in all cases except
$N=3,n_{\textrm{A}}=0$ (see below). For $N=4$ this yields $17$ and $21$
invariants, for $n_{\textrm{A}}=0$ and $n_{\textrm{A}}=1$, respectively. We
note that our count differs from that of [15], where NCGs were not considered
and the dimension of the surface was assumed to be $8N$ (from $4N$ complex
gain parameters).

Gauge Theory on a Graph.—Given the measured correlations, we wish to find a
complete and independent set of invariants. In a standard gauge theory, this
is done by fixing a base point and considering parallel transport (holonomy)
along closed loops which start and end at the base point. Such parallel
transport measures curvature and is called a Wilson loop. Wilson loops
generate all the gauge invariant quantities in a gauge theory and are the key
to constructing closure invariants. We fix a base vertex at one of the array
elements labeled $0$. We regard each array element, $a$, as a vertex in a
graph with $N$ vertices. Each element pair, $(a,b)$, is a directed edge or a
“link” in the graph, carrying the variable, $\mathbf{C}_{ab}$, a $2\times 2$
complex matrix from the $\textrm{GL}(2,\mathbb{C})$ group. This link variable
is called a “connection” and defines parallel transport from $a$ to $b$. The
gains, $\mathbf{G}_{a}$, are local (i.e., element-based) gauge transformations
by the $\textrm{GL}(2,\mathbb{C})$ group.

In standard gauge theories, such as $\mathrm{SU}(2)$ for example, the gauge
group describing $\mathbf{G}_{a}$ is unitary and Hermitean conjugation in Eq.
(1) is the same as matrix inversion. Since our gauge group is not unitary, we
have to proceed differently. So, we define a hat operator on invertible
matrices, $\widehat{\mathbf{P}}=(\mathbf{P}^{\dagger})^{-1}$ (its relevant
properties are listed in appendix I). This convenient notation lets us adapt
methods employed by [15]. We first define covariants,
$\bm{\mathcal{C}}_{\Gamma_{0}}$, as products of an even number of
$\mathbf{C}_{ab}$ matrices around a closed loop, $\Gamma_{0}$, pinned at
vertex $0$, with the even terms ‘hatted’. For example, for $\Gamma_{0}=0abc0$,
$\bm{\mathcal{C}}_{\Gamma_{0}}=\mathbf{C}_{0a}\widehat{\mathbf{C}}_{ab}\mathbf{C}_{bc}\widehat{\mathbf{C}}_{c0}$.
Under gauge transformations, covariants transform as

$\displaystyle\bm{\mathcal{C}}$
$\displaystyle\xmapsto{\mathcal{G}}\mathbf{G}_{0}\bm{\mathcal{C}}\mathbf{G}_{0}^{-1}\,.$
(3)

Note that by defining covariants pinned at vertex $0$, we have considerably
reduced the number of gains contributing to corruptions. All that remains is a
single gain $\mathbf{G}_{0}$ at the base vertex 0. Closure invariants can be
found by taking traces, $\mathcal{I}=\textrm{tr}(\bm{\mathcal{C}})$, of
covariants as in [15] or indeed in gauge theories. These invariants will be
unchanged under any gauge transformation with $\mathbf{G}_{a}$. However, the
need for an even number of links to construct covariants would seem to force
us towards rectangles. The problem now is that it is far from clear which set
will give us a complete and independent set of invariants.

Our solution to this problem is to define and use quantities called
‘advariants’, which we introduced in [17]. They are constructed by multiplying
an odd number of correlations, $\mathbf{C}_{ab}$, around a closed loop,
$\Gamma_{0}$, pinned at vertex $0$ with even terms hatted. For example, for
$\Gamma_{0}=0abcd0$,
$\bm{{\mathcal{A}}}_{\Gamma_{0}}=\mathbf{C}_{0a}\widehat{\mathbf{C}}_{ab}\mathbf{C}_{bc}\widehat{\mathbf{C}}_{cd}\mathbf{C}_{d0}$.
Under gauge transformations, advariants transform as

$\displaystyle\bm{\mathcal{A}}$
$\displaystyle\xmapsto{\mathcal{G}}\mathbf{G}_{0}\bm{\mathcal{A}}\mathbf{G}_{0}^{\dagger}\,,$
(4)

One special case of an advariant is the auto-correlation,
$\bm{\mathcal{A}}_{0}\coloneqq\mathbf{A}_{00}$, which obeys Eq. (4). The next
non-trivial example is a 3-vertex advariant defined on an elementary triangle
$\Delta_{(0ab)}\equiv(0,a,b)$,
$\bm{\mathcal{A}}_{ab}\coloneqq\mathbf{C}_{0a}\widehat{\mathbf{C}}_{ab}\mathbf{C}_{b0}$.
Note that $\bm{\mathcal{A}}_{ab}^{\dagger}$ is also an advariant. Advariants
are the building blocks for covariants. In fact, any covariant can be obtained
by multiplying an even number of such elementary advariants with even terms
hatted as described earlier. For example, a 4-vertex covariant on a closed
loop, $\Gamma_{0}=0abc0$, can be written as
$\bm{\mathcal{C}}_{\Gamma_{0}}=\bm{\mathcal{A}}_{ab}\widehat{\bm{\mathcal{A}}}_{bc}$.
Note that all these covariants transform in accordance with Eq. (3) for a
closed loop pinned at vertex 0. The traces of these covariants are the closure
invariants [15]. The key advance in this paper is that we use elementary
triangular advariants to arrive at a complete and independent set of closure
invariants, using general arguments based on Lorentz and scaling symmetries.
An advariant, $\bm{\mathcal{A}}$, can be expanded in terms of the identity
matrix, $\bm{\sigma}_{0}\coloneqq\mathbf{I}$, and Pauli matrices,
$\bm{\sigma}_{m},m=1,2,3$ as

$\displaystyle\bm{\mathcal{A}}$ $\displaystyle=z^{\mu}\bm{\sigma}_{\mu}\,,$
(5)

where, $z^{\mu}$ are complex coefficients. The $z^{\mu}$ can be thought of as
similar to complex Stokes parameters $\\{I,Q,U,V\\}$. In the case of the
autocorrelation advariant, which is Hermitean, these are real and are the
ordinary Stokes parameters of polarization optics [18].

Under gauge transformations of $\bm{\mathcal{A}}$, the action of the gauge
group, $\mathbf{G}_{0}\in\textrm{GL}(2,\mathbb{C})$, on advariants can be
split into two parts. First, the subgroup, $\mathbf{G}_{0}=\lambda\mathbf{I}$,
acts on $z^{\mu}$ by scaling,

$\displaystyle z^{\mu}$
$\displaystyle\mapsto\left|\lambda\right|^{2}z^{\mu}\,.$ (6)

Second, the subgroup $\textrm{SL}(2,\mathbb{C})$ of matrices with unit
determinant preserves
$\textrm{det}(\bm{\mathcal{A}})=I^{2}-Q^{2}-U^{2}-V^{2}=z^{\mu}\eta_{\mu\nu}z^{\nu}$,
which shows that it acts on $z^{\mu}$ by a Lorentz transformation,

$\displaystyle z^{\mu}$
$\displaystyle\mapsto\Lambda^{\mu}{}_{\nu}\,z^{\nu}\,,$ (7)

where, $\bm{\Lambda}\equiv\Lambda^{\mu}{}_{\nu}$ is a real $4\times 4$ matrix
representing a Lorentz transformation. Note that the real and imaginary parts
of ${\mathbf{z}}$ separately transform as 4-vectors. Closure invariants must
remain unchanged under transformations by both these subgroups.

Complete and independent set.— The triangular advariants (pinned at vertex 0)
can be used to build covariants for every closed loop with even links in the
graph. Each triangular advariant, $\bm{\mathcal{A}}_{ab}$, can be expanded as
$\bm{\mathcal{A}}_{ab}=z_{ab}^{\mu}\bm{\sigma}_{\mu}$ as in Eq. (5), where,
$z_{ab}^{\mu}$ are complex $4-$vectors.

We first consider only the Lorentz transformations in Eq. (7). Each triangle,
$\Delta_{(0ab)}$, gives us a complex 4-vector, $z^{\mu}_{ab}$, which can be
decomposed into two real 4-vectors. The number of triangles with one vertex at
$0$ is $N_{\Delta}=\binom{N-1}{2}$. In addition, for $n_{\textrm{A}}=1$, there
will be one more Hermitian advariant, $\bm{\mathcal{A}}_{0}$, which gives us
one more real 4-vector. Thus, we have a set of $M=2N_{\Delta}+n_{\textrm{A}}$
real, independent, 4-vectors, $\bm{y}_{m},m=1,2,\ldots M$. Invariants of the
Lorentz subgroup (Lorentz invariants), can only be functions of Minkowskian
inner products among these vectors.

We construct an independent set of closure invariants (for $N>3$) as follows:
choose any four real 4-vectors to be the basis set, say $\bm{h}_{k}={\bf
y}_{k},k=1,2,3,4$. Between these four independent basis vectors, there are
$10$ Minkowskian inner products,
$\widetilde{\mathcal{I}}_{k\ell}=\bm{h}_{k}\cdot\bm{h}_{\ell}$, with
$k,\ell=1,2,3,4$ and $\ell\geq k$. Further, the remaining $(M-4)$ 4-vectors
give us $4(M-4)$ Minkowski inner products with the four basis vectors,
$\widetilde{\mathcal{I}}_{kn}=\bm{h}_{k}\cdot\mathbf{y}_{n}$ for $n=5,\ldots
M$. Together, we have $10+4(M-4)=4N^{2}-12N+2+4\,n_{\textrm{A}}$ real
Minkowski inner products. There is no need to take any more inner products
since a 4-vector is completely specified by its components in a basis. The $M$
triangular advariants are independent and all covariants can be constructed
from these triangular advariants. This ensures that these Minkowskian inner
products are both complete and independent.

The inner products (Lorentz invariants) are still not the invariants we seek
since they acquire a scale factor from the element-based gains [Eq. (6)]. We
can obtain true invariants by forming ratios of the Lorentz invariants
$\\{\widetilde{\mathcal{I}}_{k\ell},\widetilde{\mathcal{I}}_{kn}\\}$, for
example, dividing the set by any one of them, say
$\widetilde{\mathcal{I}}_{11}$. Thus, the number of closure invariants will be
$4N^{2}-12N+1+4\,n_{\textrm{A}}$, that is, one less than the number of
independent inner products. Comparing with Eq. (2), we find that $s=1$ (one
NCG corresponding to a uniform phase across all elements) for $N\geq 4$.

The algorithmic summary for finding the closure invariants from the measured
visibilities is: 1) Choose a base element $0$ and construct the
$N_{\Delta}=\binom{N-1}{2}$ complex triangular advariants. 2) Use these to
contruct $2N_{\Delta}$ real 4-vectors. 3) If $n_{\textrm{A}}=1$, include one
more real 4-vector to get $M=2N_{\Delta}+n_{\textrm{A}}$ 4-vectors. 4) Pick
four of these 4-vectors as a basis and compute the 10 Minkowski dot products
between them. 5) Compute the Minkowski dot products between the remaining
$(M-4)$ 4-vectors and the basis vectors. 6) Divide the $10+4(M-4)$ Minkowski
dot products by any one of them to find all the closure invariants. Finding a
complete and independent set of closure invariants in polarized interferometry
is the main new result of this paper.

The case $N=3$ is special. For $n_{\textrm{A}}=1$, we get 5 invariants as
expected (from $4N^{2}-12N+1+4\,n_{\textrm{A}}$). For $n_{\textrm{A}}=0$, we
may likewise expect to get $1$ invariant. Instead, we get $2$ invariants: the
three inner products of the two 4-vectors associated with
$\bm{\mathcal{A}}_{12}$ give us two invariant ratios. The “extra” invariant
appears because there is an extra one parameter family of NCGs. These are
Lorentz transformations in the plane orthogonal to the two $4$-vectors. We
identified a uniform phase shift on all elements as an example of an NCG. Are
there others? The work of this paper and comparison with Eq. (2) clearly rules
this out in all cases but one ($N=3,n_{\textrm{A}}=0$). While this special
case is discussed in appendix II, a simple proof of this general statement can
be given: any Lorentz transformation that preserves all the visibilities also
preserves the advariants and the associated $4$-vectors. The only Lorentz
tranformation that preserves three (or more) independent vectors is the
identity. The only case where there are fewer than three independent
$4$-vectors occurs when $N=3,n_{\textrm{A}}=0$, as was noted earlier as an
example of an NCG. For $N>3$, there are no NCGs in the Lorentz subgroup. There
is just one in the scaling subgroup, a constant phase factor [17].

Comparison with earlier work.— We now return to the formulation of invariants
in terms of closure traces introduced in [15], and relate it to the framework
of Lorentz invariants constructed from 4-vectors developed in this paper. An
elementary calculation shows that

$\displaystyle\widehat{\bm{\mathcal{A}}}$
$\displaystyle=\frac{z^{0*}\bm{\sigma}_{0}-z^{1*}\bm{\sigma}_{1}-z^{2*}\bm{\sigma}_{2}-z^{3*}\bm{\sigma}_{3}}{(\bm{\mathit{z}}\cdot\bm{\mathit{z}})^{*}}\,.$
(8)

Given two advariants, $\bm{\mathcal{A}}$ and $\bm{\mathcal{A}}^{\prime}$, we
form a covariant, $\bm{\mathcal{A}}\widehat{\bm{\mathcal{A}}^{\prime}}$.
Taking the trace and using Eq. (8), we find an invariant,

$\displaystyle\frac{1}{2}\,\textrm{tr}\left(\bm{\mathcal{A}}\widehat{\bm{\mathcal{A}}^{\prime}}\right)$
$\displaystyle=\frac{\bm{\mathit{z}}\cdot\bm{\mathit{z}}^{\prime*}}{(\bm{\mathit{z}}^{\prime}\cdot\bm{\mathit{z}}^{\prime})^{*}}\,,$
(9)

which is a ratio of Minkowskian inner products. This equation provides the
link between the closure trace formulation [15] and the Minkowskian one
presented here. Appendix III discusses an interesting point that the use of
closely spaced elements not resolving the object, introduced in [15] as a
diagnostic, can be effectively extended to provide auto-correlation
information ($n_{A}=1$).

We have performed numerical tests to check the analytical theory described in
this paper. The expected number of independent invariants was confirmed by
computing the rank of the Jacobian of the numerical partial derivatives of the
invariants with respect to the components of the correlation matrices. We used
multiple realizations of the correlations, and examined the rank of the
Jacobian using its singular values. The rank agreed with our analytic results
in all cases, including for $N>4$.

In the case of $N=4,n_{\textrm{A}}=1$, a set of covariants is exhibited in
[15] (section 3.2 and appendix D). Because they did not account for the NCGs,
they expected to find only 20 real independent invariants from 10 complex
closure traces. We have numerically verified that only 18 of their 20 real
invariants are independent and further that the complete and independent set
consists of 21 real invariants when the NCGs are properly accounted for. This
illustrates the difficulty of ensuring independence and completeness within
the closure trace formalism of [15](see appendix IV). We conclude that the new
ideas and discussions in [15] are valid and valuable, but the details
regarding the independent loops need re-examination.

Discussion.— The 4-vector formalism gives us an elegant criterion for deciding
if the object has any polarization structure at all. In the absence of
polarization structure, the 4-vectors corresponding to the true correlations
have only a $0$-th component (which depends only on the Stokes total intensity
components of the correlations), which renders them all collinear. As a
result, $y_{m}^{\mu}$ are all collinear, since this property is preserved by
Lorentz transformations and scaling. The dimension of the space spanned by the
four vectors thus gives a strong statistical test for evidence of polarized
structure.

This work opens up immediate applications, and areas for further
investigation. In cutting-edge VLBI work on polarized emission, the
availability of a full set of calibration-independent constraints will provide
a valuable confirmation of derived images. Interferometry has been used, even
without imaging, to study the statistics of random fields. These studies will
also gain by a knowledge of closure invariants.

In this context, the 4-vectors introduced in this paper have the advantage
that they belong to a linear space, so that the same information can be
encoded in linear combinations, ranked by signal to noise, via singular value
decomposition. Given that direct geometric interpretation of invariants is
still being explored even in the case of co-polar measurements [8], there is
clearly much work to be done on the 4-vector approach. Finally, we anticipate
that the general concept of gauge invariance we have introduced may have wider
applicability, in other fields where general linear transformations of multi-
channel data by unknown or ill-constrained factors have corrupted the
correlations.

Acknowledgments.— We thank Dom Pesce for a correspondence clarifying the work
of [15], Supurna Sinha, Ron Ekers, and the anonymous reviewers for a careful
reading of our manuscript, and Roshni Rebecca Samuel for her rendering of Fig.
1. J. S. acknowledges support by a grant from the Simons Foundation (677895,
R.G.).

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##  I The “Hat” operator

Some of the basic properties of the hat operator are:

  * •

The order of the inverse and conjugate-transpose operations is irrelevant,
$\widehat{\mathbf{P}}=(\mathbf{P}^{-1})^{\dagger}=(\mathbf{P}^{\dagger})^{-1}$

  * •

If $\mathbf{Q}=\mathbf{P}_{1}\ldots\mathbf{P}_{j}\ldots\mathbf{P}_{n}$, is a
product of invertible matrices, $\mathbf{P}_{j}$, then
$\widehat{\mathbf{Q}}=\widehat{\mathbf{P}}_{1}\ldots\widehat{\mathbf{P}}_{j}\ldots\widehat{\mathbf{P}}_{n}$
retains the same order of multiplication of the ‘hatted’ matrices,
$\widehat{\mathbf{P}}_{j}$.

  * •

$\widehat{\mathbf{C}}_{ab}\mathbf{C}_{ba}=\mathbf{I}$ (the identity matrix)

##  II Invariants with $N=3$

As mentioned in the main text, the dimension of the stabilizer for any $N\geq
4$ is $\textrm{dim}(\mathcal{H})=1$. The stabilizer subgroup, $\mathcal{H}$,
consists of a constant phase shift applied to the signals from all the
interferometer elements, which would change $\mathbf{G}_{a}$ but not
$\mathbf{C}_{ab}$ because correlations in standard astronomical interferometry
applications depend only on the relative and not absolute phases. This means
that the stabilizer contains at least the subgroup of multiplication by
$\mathbf{G}_{a}=e^{i\theta}$ for all $a$, which is identified with $U(1)$,
since such a phase factor is simply a $1\times 1$ unitary matrix. Thus,
$\mathrm{dim}(\mathcal{H})\geq 1$. In fact, in all the cases except the one
below, $\mathrm{dim}(\mathcal{H})=1$.

$N=3$ presents an interesting degenerate case because it illustrates some
general principles behind our construction. Consider the case without auto-
correlations ($n_{\textrm{A}}=0$). There is just one triangle,
$\Delta_{(012)}$, which gives us only one complex advariant (and so, only one
complex 4-vector, $z^{\mu}_{12}$), which gives us just two real 4-vectors,
$y^{\mu}_{1}$ and $y^{\mu}_{2}$. We can form three Lorentz invariants from
Minkowski dot products,
$\tilde{\mathcal{I}}_{k\ell}=\mathbf{y}_{k}\cdot\mathbf{y}_{\ell}=y^{\mu}_{k}\eta_{\mu\nu}y^{\nu}_{\ell}$,
with $k,\ell=1,2$ and $\ell\geq k$, which will give us two closure invariants,
$\tilde{\mathcal{I}}_{kl}/\tilde{\mathcal{I}}_{11}$, after eliminating the
unknown scale factor. This is one more than that given by the formula
$4N^{2}-12N+4n_{A}+1$ that works for $N>3$.

This increase in closure invariants occurs because the stabilizer now has
$\textrm{dim}(\mathcal{H})=2$. To show this, we use the fact that there is an
extra one-parameter subgroup of Lorentz transformations in the plane
orthogonal to the two 4-vectors, $\mathbf{y}_{1}$ and $\mathbf{y}_{2}$, which
acts trivially on them (i.e., leave them unaffected).

Corresponding to these Lorentz transformations, we have a one-parameter group
of gauge transformations at the base $0$, $\mathbf{G}_{0}(s)=e^{s\mathbf{K}}$,
which leaves $\bm{\mathcal{A}}_{12}$ unchanged

$\displaystyle\mathbf{G}_{0}(s)\bm{\mathcal{A}}_{12}\mathbf{G}_{0}^{\dagger}(s)$
$\displaystyle=\bm{\mathcal{A}}_{12}\,.$ (10)

Note that the Lorentz transformations, $\mathbf{G}_{0}(s)$, belong to the
subgroup $\textrm{SL}(2,\mathbb{C})$ and have determinant unity, and so do not
affect the scale of the vectors. The question now is whether we can find a
one-parameter subgroup of gauge transformations, acting at all elements,
$\\{\mathbf{G}_{0}(s),\mathbf{G}_{1}(s),\mathbf{G}_{2}(s)\\}$, such that none
of the measured correlations is affected:

$\displaystyle\mathbf{C}_{01}$
$\displaystyle=\mathbf{G}_{0}(s)\mathbf{C}_{01}\mathbf{G}^{\dagger}_{1}(s)$
(11) $\displaystyle\mathbf{C}_{02}$
$\displaystyle=\mathbf{G}_{0}(s)\mathbf{C}_{02}\mathbf{G}^{\dagger}_{2}(s)$
(12) $\displaystyle\mathbf{C}_{12}$
$\displaystyle=\mathbf{G}_{1}(s)\mathbf{C}_{12}\mathbf{G}^{\dagger}_{2}(s)\,.$
(13)

The answer is yes: we solve Eq. (11) for $\mathbf{G}_{1}(s)$ and Eq. (12) for
$\mathbf{G}_{2}(s)$ to express them in terms of $\mathbf{G}_{0}(s)$ and the
measured correlations. This yields
$\mathbf{G}_{1}(s)=\mathbf{C}_{01}^{\dagger}\widehat{\mathbf{G}}_{0}(s)\widehat{\mathbf{C}}_{01}$,
and
$\mathbf{G}_{2}(s)=\mathbf{C}_{02}^{\dagger}\widehat{\mathbf{G}}_{0}(s)\widehat{\mathbf{C}}_{02}$.
It then follows from Eq. (10) that $\mathbf{C}_{12}$ is also unchanged by the
the one-parameter group of gauge transformations,
$\\{\mathbf{G}_{0}(s),\mathbf{G}_{1}(s),\mathbf{G}_{2}(s)\\}$. Thus, this
subgroup fixes all the measured correlations $\mathbf{C}_{ab}$, and so belongs
to the stabilizer subgroup.

This increase in the dimension of the stabilizer increases the number of
invariants by one. The two invariants can also be exhibited by taking the
trace of the covariant,
$\bm{\mathcal{A}}_{12}\widehat{\bm{\mathcal{A}}}_{12}$, and its inverse. This
is formed from the six-edged loop in which the triangle,
$\Delta_{1}\equiv\Delta_{(012)}$, is traversed twice. While the traces of this
matrix, and its inverse, are both complex, they are conjugates of each other.
So, we get two real invariants from this combination. This is also clear from
the representation of the trace in terms of inner products of 4-vectors,

$\displaystyle\mathcal{T}_{\Delta_{1}\Delta_{1}}=\frac{1}{2}\,\textrm{tr}[\bm{\mathcal{A}}_{12}\widehat{\bm{\mathcal{A}}}_{12}]$
$\displaystyle=\frac{\mathbf{z}_{12}\cdot\mathbf{z}_{12}^{*}}{(\mathbf{z}_{12}\cdot\mathbf{z}_{12})^{*}}\,.$
(14)

This exceptional behavior disappears if an auto-correlation,
$\bm{\mathcal{A}}_{0}$, is measured. This yields one more real 4-vector,
$y^{\mu}_{0}$, so that we now have three real 4-vectors
$\\{y^{\mu}_{0},y^{\mu}_{1},y^{\mu}_{2}\\}$, thereby removing the residual
Lorentz gauge freedom (present earlier in the plane orthogonal to the plane
containing the first two real 4-vectors, $\\{y^{\mu}_{0},y^{\mu}_{1}\\}$). The
number of Lorentz invariants, $\tilde{\mathcal{I}}_{k\ell}$ with
$k,\ell=0,1,2$ and $\ell\geq k$, is six. After eliminating the unknown scaling
factor by dividing by one of them, the number of closure invariants is
$N_{\mathcal{I}}=5$. In matrix language, $\mathbf{G}_{0}(s)$ was chosen to
leave the advariant $\bm{\mathcal{A}}_{12}$ unchanged. So if it left
$\bm{\mathcal{A}}_{0}$ in the case $N=3$, or indeed, for $N>3$, any other
advariant unchanged, it could only be by accident, since these are independent
measurements. In the most general case, $\mathbf{G}_{0}(s)$ will no longer be
a stabilizer when $n_{A}=1$ and $N=3$, or when $N>3$.

##  III Auto-correlations as the coincidence limit of Cross-correlations

As we have emphasised in the text, auto-correlations tend to be severely
affected by additional systematic effects (which cross-correlations eliminate)
and are therefore regarded as less reliable. However, when the angular size of
the object is small compared to the angular resolution determined by the
element spacing, we can find a way around this problem through the use of
cross-correlation following the discussion in Paper I [17].

This requires an extra element $0^{\prime}$ in close proximity with the base
element $0$. This can be provided by adding a new element, $0^{\prime}$, or by
splitting a dense array of phased elements, as in the case of the Atacama
Large Millimeter/submillimeter Array (ALMA) in the EHT/VLBI observations of
M87 and Centaurus A [19, 20], into two independent portions that are
geographically coincident and phasing them separately to act as independent
elements 0 and $0^{\prime}$. This provides us with an extra short-spacing
pair, $(0,0^{\prime})$, with separation, $D_{00^{\prime}}$, that is too short
to resolve the object ($\theta_{\textrm{obj}}\ll\lambda/D_{00^{\prime}}$),
where, $\theta_{\textrm{obj}}$ is the angular extent of the object, and
$\lambda$ is the wavelength of observation), but distant enough that the
gains, $\mathbf{G}_{0}$ and $\mathbf{G}_{0^{\prime}}$, are independent. Note
that because $\mathbf{S}_{a0^{\prime}}\approx\mathbf{S}_{a0}$, no new true
cross-correlations are provided by this additional element beyond that
provided by element 0, excepting $\mathbf{S}_{00^{\prime}}$, where,
$\mathbf{S}_{00^{\prime}}\approx\mathbf{S}_{00}$. Using the extra short-
spacing cross-correlation measured, $\mathbf{C}_{00^{\prime}}$, we can
construct an advariant that closely approximates that from an auto-correlation
as described below.

Let us construct the advariant based on a “thin” triangle
$\Delta_{(00^{\prime}a)}$, where $a\notin\\{0,0^{\prime}\\}$:
$\mathbf{C}_{00^{\prime}}\widehat{\mathbf{C}}_{0^{\prime}a}\mathbf{C}_{a0}=\mathbf{G}_{0}\mathbf{S}_{00^{\prime}}\widehat{\mathbf{S}}_{0^{\prime}a}\mathbf{S}_{a0}\mathbf{G}_{0}^{\dagger}$.
Under our assumption of closeness of elements 0 and $0^{\prime}$, this can be
written as
$\approx\mathbf{G}_{0}\mathbf{S}_{00}\widehat{\mathbf{S}}_{0a}\mathbf{S}_{a0}\mathbf{G}_{0}^{\dagger}=\mathbf{G}_{0}\mathbf{S}_{00}\mathbf{G}_{0}^{\dagger}$
since $\widehat{\mathbf{S}}_{0a}\mathbf{S}_{a0}=\mathbf{I}$. This is
effectively the same advariant that we would obtain from a direct measurement
of auto-correlation at element 0. This means the earlier discussion with
$n_{A}=1$ is applicable and we gain four invariants. The new advariant defines
a 4-vector and the four components of this 4-vector in an already established
frame give us four more real invariants. The use of extra short baselines was
proposed in [15] as an error diagnostic. We note here that they can also be
used to incorporate auto-correlations into the construction of invariants.

##  IV Connection to Closure Traces: completeness and independence

In the main paper, we chose to express the closure invariants as Minkowskian
inner products. This has the advantage that their independence and
completeness is manifest. However, for comparison with [15], we express the
same invariants in the language of closure traces. Let us single out two
triangles (labelled $\Delta_{1}\equiv\Delta_{(012)}$ and
$\Delta_{2}\equiv\Delta_{(023)}$, and their reversed forms
$\nabla_{1}\equiv\Delta_{(021)}$ and $\nabla_{2}\equiv\Delta_{(032)}$) and
construct the complex closure traces

$\displaystyle\mathcal{T}_{\Delta_{1}\Delta_{2}}$
$\displaystyle=\frac{1}{2}\,\textrm{tr}[\bm{\mathcal{A}}_{12}\widehat{\bm{\mathcal{A}}}_{23}]=\frac{\mathbf{z}_{12}\cdot\mathbf{z}_{23}^{*}}{(\mathbf{z}_{23}\cdot\mathbf{z}_{23})^{*}}\,,$
$\displaystyle\mathcal{T}_{\nabla_{1}\Delta_{2}}$
$\displaystyle=\frac{1}{2}\,\textrm{tr}[\bm{\mathcal{A}}_{12}^{\dagger}\widehat{\bm{\mathcal{A}}}_{23}]=\frac{\mathbf{z}_{12}^{*}\cdot\mathbf{z}_{23}^{*}}{(\mathbf{z}_{23}\cdot\mathbf{z}_{23})^{*}}\,,$
$\displaystyle\mathcal{T}_{\Delta_{2}\Delta_{1}}$
$\displaystyle=\frac{1}{2}\,\textrm{tr}[\bm{\mathcal{A}}_{23}\widehat{\bm{\mathcal{A}}}_{12}]=\frac{\mathbf{z}_{23}\cdot\mathbf{z}_{12}^{*}}{(\mathbf{z}_{12}\cdot\mathbf{z}_{12})^{*}}\,,$
$\displaystyle\mathcal{T}_{\nabla_{2}\Delta_{1}}$
$\displaystyle=\frac{1}{2}\,\textrm{tr}[\bm{\mathcal{A}}_{23}^{\dagger}\widehat{\bm{\mathcal{A}}}_{12}]=\frac{\mathbf{z}_{23}^{*}\cdot\mathbf{z}_{12}^{*}}{(\mathbf{z}_{12}\cdot\mathbf{z}_{12})^{*}}\,,$
$\displaystyle\mathcal{T}_{\Delta_{1}\Delta_{1}}$
$\displaystyle=\frac{1}{2}\,\textrm{tr}[\bm{\mathcal{A}}_{12}\widehat{\bm{\mathcal{A}}}_{12}]=\frac{\mathbf{z}_{12}\cdot\mathbf{z}_{12}^{*}}{(\mathbf{z}_{12}\cdot\mathbf{z}_{12})^{*}}\,,$
$\displaystyle\mathcal{T}_{\Delta_{2}\Delta_{2}}$
$\displaystyle=\frac{1}{2}\,\textrm{tr}[\bm{\mathcal{A}}_{23}\widehat{\bm{\mathcal{A}}}_{23}]=\frac{\mathbf{z}_{23}\cdot\mathbf{z}_{23}^{*}}{(\mathbf{z}_{23}\cdot\mathbf{z}_{23})^{*}}\,.$
(15)

The other pairwise advariant combinations not written here can be shown to be
dependent on the above.

It would appear that we have twelve real invariants from these six complex
closure traces. However, they have further dependencies satisfying the
following three real relations:

$\displaystyle\left|\frac{\mathcal{T}_{\nabla_{1}\Delta_{2}}\mathcal{T}_{\Delta_{2}\Delta_{1}}}{\mathcal{T}_{\Delta_{1}\Delta_{2}}\mathcal{T}_{\nabla_{2}\Delta_{1}}}\right|$
$\displaystyle=1\,,$ (16)
$\displaystyle\frac{\mathcal{T}_{\Delta_{1}\Delta_{2}}^{*}\mathcal{T}_{\nabla_{2}\Delta_{1}}\mathcal{T}_{\Delta_{2}\Delta_{2}}}{\mathcal{T}_{\nabla_{1}\Delta_{2}}\mathcal{T}_{\Delta_{2}\Delta_{1}}\mathcal{T}_{\Delta_{2}\Delta_{2}}^{*}}$
$\displaystyle=1\,,$ (17)
$\displaystyle\textrm{and}\quad\frac{\mathcal{T}_{\nabla_{1}\Delta_{2}}\mathcal{T}_{\Delta_{2}\Delta_{1}}^{*}\mathcal{T}_{\Delta_{1}\Delta_{1}}}{\mathcal{T}_{\Delta_{1}\Delta_{2}}\mathcal{T}_{\nabla_{2}\Delta_{1}}\mathcal{T}_{\Delta_{1}\Delta_{1}}^{*}}$
$\displaystyle=1\,.$ (18)

This reduces the number of real invariants to 9, made from the advariants
$\bm{\mathcal{A}}_{12}$ and $\bm{\mathcal{A}}_{23}$ alone. In addition, each
of the remaining triangles, $\Delta_{k}$ with $k\geq 3$, contributes the
following four complex invariants constructed by taking the traces of the
products of each $\bm{\mathcal{A}}_{\Delta_{k}}$ with
$\bm{\mathcal{A}}_{\Delta_{1}}$ and $\bm{\mathcal{A}}_{\Delta_{2}}$, and their
Hermitian adjoints,

$\displaystyle\mathcal{T}_{\Delta_{1}\Delta_{k}}=\frac{1}{2}\,\textrm{tr}[\bm{\mathcal{A}}_{\Delta_{1}}\widehat{\bm{\mathcal{A}}}_{\Delta_{k}}]$
$\displaystyle\,,\,\mathcal{T}_{\Delta_{2}\Delta_{k}}=\frac{1}{2}\,\textrm{tr}[\bm{\mathcal{A}}_{\Delta_{2}}\widehat{\bm{\mathcal{A}}}_{\Delta_{k}}]\,,$
$\displaystyle\mathcal{T}_{\nabla_{1}\Delta_{k}}=\frac{1}{2}\,\textrm{tr}[\bm{\mathcal{A}}_{\Delta_{1}}^{\dagger}\widehat{\bm{\mathcal{A}}}_{\Delta_{k}}]$
$\displaystyle\,,\,\mathcal{T}_{\nabla_{2}\Delta_{k}}=\frac{1}{2}\,\textrm{tr}[\bm{\mathcal{A}}_{\Delta_{2}}^{\dagger}\widehat{\bm{\mathcal{A}}}_{\Delta_{k}}]\,.$
(19)

As earlier, we can express these in terms of ratios of Lorentz-invariant
Minkowskian inner products, and in this case it is easy to see that the four
complex invariants are indeed independent. This gives us eight real invariants
per additional triangle, $\Delta_{k}$, for $3\leq k\leq(N-1)(N-2)/2$, in full
agreement with the counting based purely on Lorentz invariants. Also, note
that the argument in the main text based on Lorentz invariants is considerably
simpler than the one described above.

##  V Holonomy and the Choice of Base Vertex

In gauge theory, it is usual to talk about the holonomy group at a point, $P$,
which gives us the parallel transport along loops $\Gamma$ pinned at $P$. One
can multiply two loops by traversing them in succession and the holonomy of
the product of the loops is the product of the holonomy of the individual
loops. In unitary lattice gauge theories (and more generally gauge theory on a
graph), there are elementary plaquettes and all loops can be broken into these
elementary plaquettes, which are both independent and complete. In our
problem, the gauge group was not unitary, and the even holonomies were
quadrilateral and therefore not independent. We got around this difficulty by
using triangular advariants which are both independent and complete. The
advariants are not themselves holonomies, but can be used to construct all
holonomies.

The holonomy group depends on the base vertex, but only changes by
conjugation. So, the choice of the base vertex is not important from the
mathematical point of view, and we have arbitrarily singled out one array
element (indexed 0) as the base vertex. In practice, the choice of the base
vertex may be guided by other considerations like the quality of data received
at different elements. A final remark about nomenclature: the word “holonomy”
which is used in the mathematical literature is known to physicists as a
“Wilson loop”. In the geometric phase literature, it is sometimes referred to
as an “anholonomy”, following Michael Berry’s just observation [21, 9] that we
are really describing a failure of integrability and therefore a lack of
“wholeness”. This usage is consistent with “holonomic and anholonomic
constraints” of classical mechanics.