It is known that canonical commutation relations do not fix the form of momentum operator. That means that if canonical commutation relations (CCR) are given by
$$[\hat{x}^i,\hat{p}_j]~=~i\hbar~\delta^i_j~ {\bf 1}$$
they can be satisfied by the following choice of momentum operators:
$p_x = -ih\frac{∂}{∂x}+\frac{∂f}{∂x}$
$p_y = -ih\frac{∂}{∂y}+\frac{∂f}{∂y}$
$p_z = -ih\frac{∂}{∂z}+\frac{∂f}{∂z}$
where $f(x,y,z)$ - arbitrary function.
On the other hand, for any choice of $f(x,y,z)$ momentum operators can be transformed to their most frequently used form $(-ih\frac{∂}{∂x})$ (etc for $y$ and $z$) by the following transformation of the wave function $\psi$ and operators $p$:
$\psi'=e^{-\frac{i}{h}f(x,y,z)}\psi$
$p^{'}_x =e^{-\frac{i}{h}f(x,y,z)}p_x e^{+\frac{i}{h}f(x,y,z)}=-ih\frac{∂}{∂x}$
Hence, we obtain $U(1)$ gauge transformation using only canonical commutation relations for momentum and position operators.
Does this mean that $U(1)$ gauge invariance corresponds to conservation of momentum rather than to conservation of electric charge?
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