# 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?

Many thanks in advance.

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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).