# Does the canonical commutation relation give a unique solution for the momentum operator? [duplicate]

So lets say we are in a 1d system and in the position basis just for simplicity. The CCR is: $$[x,p]=i$$ and the momentum operator is $$-i\partial_x$$. Is this solution unique or are there other operators which could satisfy the commutation relation.

Similarly, in QFT does the commutation relation $$[\phi(x),\pi(y)]=i\delta(x-y)$$ uniquely specify that (in the eigenbasis of $$\phi$$) $$\pi(x)=-i\frac{\delta}{\delta \phi(x)}$$?

Obviously these are a solution to the algebra, but are they unique?

In my experience this commutation relation usually has a different sign: $$[x,p]=i$$, but this is a matter of definition. Derivative $$\hat{p}=-i\partial_x$$ is a possible representation of operator $$\hat{p}$$ that satisfies this equation, but not the only possible and not necessarily the best. Let us take, e.g., two Bose operators $$\hat{a},\hat{a}^\dagger$$ that have matrix representation $$\langle m|\hat{a}|n\rangle = \sqrt{n}\delta_{m, n-1}, \langle m|\hat{a}^\dagger|n\rangle = \sqrt{n+1}\delta_{m, n+1},$$ and satisfy commutation relation $$[\hat{a},\hat{a}^\dagger]=1$$. We can now construct operators $$\hat{x}=\frac{\hat{a}+\hat{a}^\dagger}{\sqrt{2}}, \hat{p}=i\frac{\hat{a}-\hat{a}^\dagger}{\sqrt{2}},$$ which satisfy the required commutation relation $$[\hat{x},\hat{p}]=i.$$ The two representations are; of course, intimately related - one could re-express $$\hat{a},\hat{a}^\dagger$$ in terms of $$x$$, $$\hat{p}=-i\partial_x$$, but one can see this as another solution satisfying this commutation relation.
• I get that you can do it in other representations. I was really asking that if we stick to the eigenbasis of $x$, is it unique? – Toby Peterken May 3 '20 at 18:03