Symmetries, Generators, Commutators and Observables

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?

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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.

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What might be some typical additional conditions? How does it relate to the example perhaps? Many thanks! – Edward Hughes Aug 16 '12 at 8:46
I don't know; each context is different. Maybe the source doesn't make use of the uniqueness but just needs some solution. Then sloppiness in the writeup is a sufficient explanation. – Arnold Neumaier Aug 16 '12 at 8:50
Okay thanks. So we have an infinity of possible conserved quantities then? – Edward Hughes Aug 16 '12 at 9:03
There is always an infinity of conserved quantities since if $A$ and $Q$ are conserved then any function of $A$ and $Q$ is also conserved. One is therefore interested in independent conserved quantities, or in additively conserved quantities. – Arnold Neumaier Aug 16 '12 at 9:23

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).

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Very many thanks for this. Do you know of any references where I could read up on this irreducible representation stuff, because I haven't come across it before? Obviously I could look at Wikipedia, but a particular reference with QM in mind would be better! – Edward Hughes Aug 15 '12 at 21:30
No problem. For basic representation theory I like Fulton and Harris' "First Course in Representation Theory". For physical applications, the representation theory of finite groups is a bit of a detour though. For stuff focused on Lie gadgets I like Kirillov's book, which also has the advantage of being freely available online. As for stuff with a focus on quantum, I'm not really sure. I know Peter Woit has written some things about quantum mechanics and representation theory, but I think it is more about "geometric quantization" and would probably not be useful to you. – Ryan Thorngren Aug 15 '12 at 21:37
Actually, I think the greatest reference (or reading guide, perhaps) is John Baez's This Weeks Finds. He talks about representation theory, quantum mechanics, and much much more in a very informal (but informative) style. – Ryan Thorngren Aug 15 '12 at 21:40