[转载]What’s a Gauge?

From: Terence Tao’s blog: What’s a Gauge.

Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth. But the underlying concept is really quite simple: a gauge is nothing more than a “coordinate system” that varies depending on one’s “location” with respect to some “base space” or “parameter space”, a gauge transform is a change of coordinates applied to each such location, and a gauge theory is a model for some physical or mathematical system to which gauge transforms can be applied (and is typically gauge invariant, in that all physically meaningful quantities are left unchanged (or transform naturally) under gauge transformations). By fixing a gauge (thus breaking or spending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the tractability of the resulting problem can be heavily dependent on the choice of gauge that one fixed. Deciding exactly how to fix a gauge (or whether one should spend the gauge symmetry at all) is a key question in the analysis of gauge theories, and one that often requires the input of geometric ideas and intuition into that analysis. Continue Reading

Connections and Curvatures on Vector Bundles

Let $E\to M$ be a smooth complex vector bundle over a smooth compact manifold $M$. Denote $\Omega^\cdot(M;E):=\Gamma(\Lambda^\cdot(T^\ast M)\otimes E)$ be the space of smooth sections of the tensor product vector bundle $\Lambda^\cdot(T^\ast M)\otimes E$.

A connection on $E$ is an extension of exterior differential operator $\rd$ to include the coefficient $E$.

定义 1. A connection $\nabla^E$ on $E$ is a $C^\infty(M)$-linear operator from $\Gamma(E)$ to $\Omega^1(M;E)$ such that for any $f\in C^\infty(M)$ and $s\in\Gamma(E)$, satisfy the Leibniz rule
\[
\nabla^E(fs)=(\rd f)\otimes s+f\nabla^Es.
\]
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