Why are commutators important? I'm an undergrad, and I've spent over a year studying Operator Theory, and a large part of that time learning about commutators of operators on Hilbert spaces. If $A,B$ are operators on a Hilbert space $\mathcal H$, their commutator $[A,B]$ is defined by $$[A,B] := AB- BA$$
However, I'm still not very clear about the motivation behind studying commutators of operators (especially bounded linear operators) on Hilbert spaces.

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*Are there any significant/deep/profound results about or applications of commutators that I should know of?

*Are there valuable connections to other areas in mathematics, or even physics?

Feel free to post whatever you feel is relevant in the answers or comments! I'm interested to gather different perspectives. Thank you!
 A: Don't have enough points to comment, but I can give some pretty standard examples from physics.
In quantum mechanics, one is very interested in the noncommutativity of operators on Hilbert space as they essential manifest uncertainty principles. Commutators let one test if it is possible to simultaneously know with a given certainty (standard deviation) the values of two quantum mechanical observables; does measuring one quantity before the other affect the statistical ensemble?
So for example, an axiom called the canonical commutator relation shows the difference between measuring position (with operator $\hat{x}$) then momentum (with operator $\hat{p}$) vs momentum then position in a quantum state $|\Psi>$ - an element of a Hilbert space.
$$[\hat{x},\hat{p}] = i \hbar$$
This axiom was first given by Max Born and Pascual Jordan (1925). That directly leads to the famous Heisenberg uncertainty principle as for any operators A and B on H,
$$\sigma_A \sigma_B \geq \big|\frac{1}{2i} <[A,B] > \big|$$
where angled brackets are the expectation and the sigmas are standard deviations. See a proof an discussion here.
On the other hand, when two operators commute, it is possible to simultaneously know the values they extract from quantum states. For example one is often interested in commuting things with the Hamiltonian (operator that gives energy). When modeling the Hydrogen atom, you find that the operator for angular momentum (of the electron) commutes with the Hamiltonian of the setup, so it is possible to know the energy and angular momentum of an electron in the Hydrogen atom at the same time; these operators (by axiom) extract physical values by eigenvalue equations. So the system {$\hat{H} |\Psi> = E |\Psi>$ and $\hat{L} |\Psi> = l |\Psi>$} can be satisfied for eigenvalues $E$ and $l$.
These compatible observables are critical in how we understand chemistry - they determine orbitals (along with another number called spin).
So okay. QM uses operators to make physical observations essentially. Commutators tell us when it is impossible (and how much so) to know two observables at once. But why should they be bounded and linear.
Linearity is a generally desirable in physics. The ODEs that arise from these operators are much simpler to handle and they introduce the physical concept of superposition. Linearity is in some sense physically natural; if some set of waves satisfies a mathematical condition of a wave equation, why shouldn't their sum also be a valid wave? That brings in Fourier analysis deeply into the fold of QM and particle physics in general.
The bounding of these operators is also nice. One defines the expectation of a quantum operator as $<\Psi|\hat{O} |\Psi>$ (when the quantum state is normalized, i.e. the inner product $(\Psi,\Psi) =1$). One would want the observable quantities to be bounded. For example, a lower bound on such an expectation condition tells you there must be energy in the vacuum.
