Symmetries, Generators, Commutators and Observables

Question

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

$$g_a(A)=af(A)+\textrm{O}(a^2)$$

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.

Answer 1

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 in the context of your source.

Answer 2

If a group has an irreducible representation on some vector space, then Schur's lemma says the only operators on that vector space which commute with every element of the group are the scalar operators. From this, If A and B have the same commutation relations with every element of the group, then $A-B$ commutes with everything, and thus must be a scalar. Therefore, the commutation relations determine everything up to the addition of a scalar (ie. up to central extension).