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I heard someone say that the definition of a gauge transformation is any formal, systematic transformation of the potentials that leaves the fields invariant. What is the definition of a potential in the context of fields (such as QFT)? If possible, in layman's terms.

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Note that the notion of gauge theories is much more general than, say, Yang-Mills theory. There is of course a long list of gauge theories that have gauge potentials -- e.g. in the case of Yang-Mills theory, the $A^a_{\mu}(x)$ field is the gauge potential -- but it is not a general requirement for a gauge theory.

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The non-derivative terms of a field $\phi$ in the Lagrangian are often (not always) collectively called its "potential" $V(\phi)$. This is because in field theory, the field $\phi$ is the analog of coordinates $q$ particle mechanics, and there potential functions usually contain powers of $q$.In the context of electromagnetism, the field $A^\mu$ itself is a potential, called electromagnetic four-potential $A^\mu=(\phi/c,\textbf{A})$.

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