Take the 2-minute tour ×  I'm learning about generators and
conservation laws and have derived the equation (1)
$$[Q,A]=-i\hbar f(A)$$  which is satisfied by the observable
generator $Q$ for a transformation group with elements of form
The lecture notes I'm reading say that this equation (1) defines
$Q$ provided we know $f(A)$ for all observables $A$. Why is this
true mathematically? And what does all observables mean?  The
example in the notes applies it to transformations along the
$k-$axis for a system of $r$ particles, obtaining
$$[Q,\hat{x}_i^r]=-i\hbar\delta_{ik}$$ $$[Q,\hat{p}_i^r]=0$$  It
then states that $Q = \hat{P}_k=\sum_r \hat{p}_k^r$. This is
obviously a solution, but do I know that it's the only one?
Many thanks in advance.  share|improve this question add comment
2 Answers  up vote 3 down vote accepted  The statement is not
correct, as $Q'=Q+g(A)$ satisfies (for an arbitrary function
$g$) the same commutation relation. Thus you need to pay
attention to the additional conditions posited i