Consider the covariant derivative ##D_{\mu}=\partial_{\mu}+ieA_{\mu}## of scalar QED.(adsbygoogle = window.adsbygoogle || []).push({});

I understand that ##D_{\mu}\phi## is invariant under the simultaneous phase rotation ##\phi \rightarrow e^{i\Lambda}\phi## of the field ##\phi## and the gauge transformation ##A_{\mu}\rightarrow A_{\mu}+\frac{1}{e}(\partial_{\mu}\Lambda)## of the vector potential ##A_{\mu}##.

I was wondering if the phase rotation and the gauge transformation are related in any way? Does the gauge transformation necessarily lead to the phase rotation of the field?

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# A Gauge invariance and covariant derivative

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