共极干涉中的不变量：阿贝尔规范理论
Invariants in Co-polar Interferometry: an Abelian Gauge Theory
摘要：
$N$ -元件干涉仪测量阵列元件对之间的相关性。与阵列元素之间的闭环相关联的闭合不变量不受这些测量中发生的基于元素的乘法（“局部”）损坏的影响。直到最近，人们还不清楚如何通过分析确定一组完整的独立不变量。我们认为，当地的，基于元素的腐败共极相关规范变换属于规范组$\textrm{GL}(1,\mathbb{C})$。封闭量自然是规范不变的。我们用它来提供一个简单而有效的形式主义，并确定完整的一套独立的封闭不变量，从共极干涉相关性，只使用数量定义的$(N-1)(N-2)/2$小学和独立的三角形循环。在天文干涉测量中熟悉的$(N-1)(N-2)/2$闭合相位和$N(N-3)/2$闭合振幅（总计$N^2-3N+1$实不变量）自然地从这种形式主义中出现，它统一了迄今为止需要单独处理的问题。我们不需要自相关性，但如果可靠地测量，可以很容易地包括它们。这个统一的观点阐明了与噪声和对象模型参数的推断有关的问题。它还允许我们将光学中与潘查拉特南相位相关的平行输运规则扩展到振幅。这里提出的框架扩展到$\textrm{GL}(2,\mathbb{C}$）中提出的全极化干涉测量的配套文件，概括和澄清早期的工作。我们的研究结果与最先进的共极和全极化甚长基线干涉测量有关，以确定M87，半人马座~A和银河系中心黑洞事件视界附近的特征。
英文摘要：
An $N$-element interferometer measures correlations among pairs of array elements. Closure invariants associated with closed loops among array elements are immune to multiplicative, element-based ("local") corruptions that occur in these measurements. Till recently, 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 $\textrm{GL}(1,\mathbb{C})$. Closure quantities are then naturally gauge invariant. We use this to provide a simple and effective formalism, and identify the complete set of independent closure invariants from co-polar interferometric correlations using only quantities defined on $(N-1)(N-2)/2$ 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. We do not require auto-correlations, but can easily include them if reliably measured. This unified view clarifies issues relating to noise and inference of object model parameters. It also allows us to extend the rule of parallel transport associated with Pancharatnam phase in optics to apply to amplitudes as well. The framework presented here extends to $\textrm{GL}(2,\mathbb{C}$) for full polarimetric interferometry as presented in a companion paper, which generalizes and clarifies earlier work. Our findings are relevant to state of the art 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.
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