Abstract
An $N$-element interferometer measures correlations between pairs of array
elements. Closure invariants associated with closed loops among array elements
are immune to multiplicative, local, element-based corruptions that occur in
these measurements. Till now, it has been unclear how a complete set of
independent invariants can be analytically determined. We view the local,
element-based corruptions in co-polar correlations as gauge tranformations
belonging to the gauge group $GL(1,C)$. Closure quantities
are then naturally gauge invariant. Using an Abelian
$GL(1,C)$ gauge theory, we provide a simple and effective
formalism to isolate the complete set of independent closure invariants from
co-polar interferometric correlations only using quantities defined on the
$(N-1)(N-2)/2$ elementary and independent triangular loops. The $(N-1)(N-2)/2$
closure phases and $N(N-3)/2$ closure amplitudes (totaling $N^2-3N+1$ real
invariants), familiar in astronomical interferometry, naturally emerge from
this formalism, which unifies what has required separate treatments until now.
Our formalism does not require auto-correlations, but can easily include them
if reliably measured, including potentially from cross-correlation between two
short-spaced elements. The gauge theory framework presented here extends to
$GL(2,C$) for full polarimetric interferometry presented in a
companion paper, which generalizes and clarifies earlier work. Our findings can
be relevant to cutting-edge co-polar and full polarimetric very long baseline
interferometry measurements to determine features very near the event horizons
of blackholes at the centers of M87, Centaurus~A, and the Milky Way.
Description
Invariants in Co-polar Interferometry: an Abelian Gauge Theory
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