View source for Gauge invariance

===Path integrals===

Following Feynman (Feynman 1948), quantum mechanics can be alternatively formulated in terms of [[path integral]]s.
In this formalism, the matrix elements of the quantum evolution operator <math>\mathcal{U}(t'',t') </math> between times <math> t'</math> and <math> t''</math> are given by 
a sum over all possible (''classical'' and ''non-classical'')  trajectories <math>\mathbf{q}(t)</math> (''paths''), which in the simplest cases can be written as 
:<math>\label{magpath}
\langle \mathbf{q}''  \left| \mathcal{U}(t'',t')   \right| \mathbf{q}'
\rangle   = \int \left [  \mathrm{d} \mathbf{q} (t) \right] \exp\left({i 
\over  \hbar}\mathcal{S} (\mathbf{q} )\right) \quad \hbox{with boundary conditions}\quad \mathbf{q}(t')=\mathbf{q}' , \  \mathbf{q}(t'')=\mathbf{q}'', </math>
 
where   <math> \mathcal{S} (\mathbf{q} ) </math>  is the classical action defined in \eqref{action}.
  
This formulation of quantum mechanics actually explains why equations of motion in classical mechanics can be derived from a variational principle in the form \eqref{euler-lagrange}. 
In the classical limit, that is, when the typical classical action is large with respect to  <math>\hbar\ ,</math> the path integral can be approximated by using the  ''stationary phase method''. The sum over paths is thus dominated by paths that leave the action stationary: the  ''classical paths'' that satisfy \eqref{euler-lagrange}. This property generalizes to relativistic quantum field theory.

In the case of a spinless non-relativistic particle in an electric and magnetic fields, the Lagrangian \eqref{LagrangianEB}, under a gauge transformation changes by a total time derivative, \eqref{LEBtrans}. The action changes then by boundary terms and, correspondingly, the evolution operator transforms as
:<math>\langle \mathbf{q}''  \left|\mathcal{U}(t'',t')   \right| \mathbf{q}'\rangle\mapsto\, U(t'',\mathbf{q''})\,\langle \mathbf{q}''  \left|\mathcal{U}(t'',t')   \right| \mathbf{q}'\rangle \, U^\dagger(t',\mathbf{q}')</math>
which is consistent with the transformation \eqref{tgaupsi} of the wave function.