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The Hamiltonian for a charged particle in electromagnetic field is invariant under the gauge transformation \[\vec A \longrightarrow \vec A^\prime = \vec A -\nabla \Lambda\]if the canonical momentum is taken to transform as \[\vec p \longrightarrow \vec p^\prime -(e/c) \nabla \Lambda.\]
$\newcommand{\Prime}{{^\prime}}\newcommand{\Label}[1]{\label{#1}}$
The electric and magnetic fields, and hence the Lorentz force on a charged particle and the equations of motion do not change under a gauge transformation of the vector potential
\begin{equation} \vec A \longrightarrow \vec A\Prime = \vec A - \nabla \Lambda. \Label{EQ06} .\end{equation}
Under transformation \eqref{EQ06}, the Lagrangian changes by a total time derivative and, hence the action remains invariant.
In the phase space formulation the invariance of Hamilton's equations is ensured by assigning the transformation \begin{equation}\Label{EQ07} \vec p \longrightarrow \vec p\Prime -(e/c)\nabla \Lambda \end{equation} to the canonical momentum.
Note that the Hamiltonian of a charged particle in electric and magnetic field is
\begin{equation}
H = \frac{(\vec p-(e/c)\vec A)^2}{2m} + e\phi,
\end{equation}
involves the the momentum and vector potential in a combination \((\vec p-(e/c)\vec A)\) which is invariant under the gauge transformations \eqref{EQ06} and \eqref{EQ07}. Thus it follows that the Hamiltonian remains unchanged under simultaneous transformations \eqref{EQ06} and \eqref{EQ07}.
In fact all observable quantities must be invariant under gauge transformations. Since the canonical momentum \(\vec p\) is not invariant under a gauge transformation, it not a physically observable quantity.
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4727:Diamond Point
