Can $AB-BA=I$ hold if both $A$ and $B$ are operators on an infinitely-dimensional vector space over $\mathbb C$?

Question

Of course, it can't hold if operators are over finite-dimensional spaces, as is evident from trace considerations. Can it be true for infinite-dimensional spaces? I think not, but I don't see how we can argue in this case.

Answer 1

Yes, there is even a typical example. If $V$ is the space of all polynomials on $\mathbb C$, and $D,x:V\to V$ are defined by: $$(Df)(x)=\frac{df}{dx}\\(xf)(x)=xf(x).$$ Then $Dx-xD=I$. You could also take $V$ to be all meromorphic functions over $\mathbb C$, or the space of power series over $\mathbb C$.

Such pairs of operators actually are the underlying explanation for the Heisenberg Uncertainty Principle.

Answer 2

Let $V = \mathbb{K}[x]$ be the space of polynomials in one variable with coefficients from your ground field $\mathbb{K}$, and consider the two operators $\frac{d}{dx}$ (formal differentiation of polynomials) and $x$ (multiplication of polynomials by $x$).

Answer 3

The canonical commutation relations for position and momentum operators in quantum mechanics is $$[\hat x,\hat p]=i \hbar.$$ Here the vector space is the Hilbert space over $\mathbb C$.