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Let $q_i,p_j$ be canonically conjugate operators for $i,j=1,2,\ldots,n$ that satisfy the relation $[p_i,q_j]=c\delta_{ij}$ where $c$ is constant and $[\cdot,\cdot]$ is a commutator.

What sort of unitary transformation of $q_i$ in the form $q_i\mapsto q_i+F_i(p_j)$ is such that the relations are invariant?

I believe if $F$ is a power series of $p_j$ then that should be the case because we can work term by term. I would like to know if there is looser condition possible. And if not, is there a way to show that $F$ has to be some form of power series?


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