GaugeEquivalenceClass - crowlogic/arb4j GitHub Wiki

Gauge Equivalence: Connections $A$ and $A'$ on a principal bundle over a manifold are gauge equivalent if there exists a gauge transformation $g$ such that $A' = gAg^{-1} + g dg^{-1}$, where $g: M \to G$ is a map from the manifold $M$ to the gauge group $G$.

Probability Measure on Gauge Equivalence Classes: The document aims to define a probability measure on the space of gauge equivalence classes. This involves constructing a measure space where each point represents an equivalence class of connections, modulo gauge transformations.
Gaussian Measures for Abelian Groups: For the abelian group $U(1)$, the document defines a Gaussian measure dependent on real parameters $m > 0$ and $c \neq 0$, leading to a non-standard quantization of the Yang-Mills Hamiltonian.

Space of Connections $\mathcal{D}$: The space of all connections on a principal bundle, which is an infinite-dimensional manifold.
Gauge Group $\mathcal{K}$: The group of all gauge transformations acting on $\mathcal{D}$. The action of $\mathcal{K}$ on $\mathcal{D}$ is given by $g \times A \mapsto gAg^{-1} + gdg^{-1}$, leading to the quotient space $\mathcal{D}/\mathcal{K}$ representing gauge equivalence classes.
Quotient Space $\mathcal{D}/\mathcal{K}$: The space of gauge equivalence classes, which is the fundamental object for defining the probability measure. It's a challenging object due to its infinite-dimensional nature and the complexity of gauge orbits.

Gauge Orbit of a Connection: The set of all connections that can be reached from a given connection $A$ by gauge transformations, forming a gauge equivalence class.
Riemannian Metric on $\mathcal{D}/\mathcal{K}$: The document discusses inducing a Riemannian metric on the space of gauge equivalence classes, which is crucial for defining and analyzing the measure.
Hamiltonian Reduction: The process of reducing the Yang-Mills Hamiltonian to a form suitable for quantization, involving the gauge equivalence classes and their properties.