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The following (from notes for a lecture I was preparing) might help explain the context: The first gauge theory was Hermann Weyl's extension of Einstein's general theory of relativity with a parallel transport that can change the scale or 'gauge' of lengths of the transported vector. About this one can read in P. G. Bergman's book on Relativity. The Hamiltonian formulation of electrodynamics, and in particular, the replacement of \(\vec{p}\) by \(\vec{p}-e\vec{A}/c \) was given by Larmor in his book "Aether and Matter", Cambridge (1900). [quoted by Pauli in ''General Principles of Quantum Mechanics" , Section 4. (Tr. by P. Achuthan and K. Venkatesan of 1958 German edition) Allied, New Delhi 1980.] In quantum mechanics the 'canonical momentum' \(\vec{p}-e\vec{A}/c\) becomes \(-i\hbar[\nabla-ie\vec{A}/(\hbar c)]\). The gauge invariance of the Schrodinger theory under \(\vec{A}\to \vec{A}+\nabla f\) and \(\phi\to \phi-(e/c)\frac{\partial f}{\partial t} \) when \(\Psi\) is changed by a phase was first given by V. Fock (1927). The analogy of this group of transformations to the Weyl theory on gravitation and electricity was pointed out by F. London (1927). The connection of this group to charge conservation was pointed out by Weyl while writing variational principle for the wave equation. [See Pauli as above.]
