Why do we care about the canonical commutation relations?

Question

Suppose $\hat{x}$ and $\hat{p}$ are the position and momentum operators, it can be shown that $$[\hat{x}, \hat{p}] = i\hbar\mathbb{I}.$$ The Stone-von Neumann theorem tells us that that the above is unique up to unitary equivalence.

I am unclear on the significance of the canonical commutation relations shown above. My current interpretation of commutators is, informally speaking, that they measure the extent to which two operators commute. What further information does the canonical commutation relation give us, and why is its uniqueness up to unitary equivalence such a big deal?

Answer 1

There's more to it, and the deeper content it encodes is related to symmetries and their associated conserved quantities. Let us start with the classical theory, to see that this is indeed already present there. In Classical Mechanics we may formulate our theory in the Hamiltonian Formalism. In that case we have a phase space $(\Gamma,\Omega)$ where $\Gamma$ is a space, which in basic mechanics courses is usually described as the space of pairs $(q^i,p_i)$ of position and momenta, and where $\Omega$ is an object called sympletic form.

The sympletic form gives rise to an operation among functions on $\Gamma$ called the Poisson bracket $\{,\}$. The Poisson bracket between position and momenta obey $$\{q^i,p_j\}=\delta^i_{\phantom i j}\tag{1}\label{ccr}.$$

Now, you might be aware of a result known as Noether's theorem which puts in correspondence symmetries and conservation laws. In the Hamiltonian Formalism it can be phrased as follows. For a given symmetry we have a function in $\Gamma$, called its Hamiltonian charge $Q$, which has the property that $$\{Q,f\}=-\delta_Q f\tag{2}$$

where $\delta_Q$ is the variation of the observable according to the symmetry corresponding to $Q$.

Now let us consider translations. Consider a translation by $\epsilon^i$ so that the coordinates get transformed as $q^i\to q^i+\epsilon^i$. We will have $\delta q^i = \epsilon^i$. In that regard, observe that if we define $Q = \epsilon^i p_i$ we have $$\{ Q,q^i\}=\epsilon^j\{p_j,q^i\}=-\epsilon^i = -\delta_Q q^i\tag{3}.$$

Observe that (1) has been used in the second equality. What this tells is that (1) is the statement that momentum is the generator of translations, or else that momentum is the Hamiltonian charge associated to translations. In particular, momentum in the $i$-th direction generates translations in the $i$-th direction, that is the content of (1).

This then naturally generalizes to Quantum Mechanics. And it is not so surprising that it happens, since we know that the correspondence principle gives the quantization rule $[] \leftrightarrow i\{\}$. In that setting, the Canonical Commutation Relations are just saying that momentum should be the generator of translations.

Obviously, the whole analysis of symmetries that I have outlined above in Classical Mechanics can be made in a self-contained manner in Quantum Mechanics. I only did it in Classical Mechanics to show you that there is a classical version of the story, which may be easier to understand first.

In summary, commutation relations often encode symmetry statements and their associated charges, and the CCR is just one example of that.

Answer 2

If you want your mathematical description to be able to describe momentum and position, it must respect these relations. This is a fundamental property of what position and momentum are, and hence any possible description you can come up with should respect that. It can also be interpreted as meaning that momentum is the generator of translations, which ties the notion of position and momentum together by specifying that momentum is a quantity that is conserved when the system has symmetry under shifting the positions.

Uniqueness up to unitary equivalence is a big deal because, otherwise, you would be able to have two different descriptions of momentum and position. You would be able to come up with two different Hilbert spaces that are not related to each other and still are, allegedly, descriptions of the same phenomena. How do you know which one is correct? Is one of them preferred? This leads to inconveniences, at the very least, and hence it is important that there is uniqueness of representation.

When uniqueness fails (bonus section)

It is interesting to point out that the Stone–von Neumann theorem fails for systems with infinitely many degrees of freedom, which arise in field theory. In those situations, you have unequivalent choices of Hilbert space. Often, a "preferred" Hilbert space is picked out by choosing a preferred observer. In flat spacetime, this choice is done implicitly by picking the Hilbert space corresponding to what would be described by inertial observers, but other observers could associate different Hilbert spaces. This means there are states that one observer considers that are not considered by other observers, for example. It does happen, however, that each Hilbert space can approximate any other one arbitrarily well, in a well-defined sense. Still, due to this inconvenience in picking a preferred Hilbert space and due to other reasons, some physicists prefer to formulate these sorts of systems in an entirely algebraic manner. One gets rid of the Hilbert space and focuses only on the properties of the observables between themselves (like the commutation relations). They constitute an algebra (i.e., a vector space with a product between vectors) and states are now understood as linear functionals on this algebra.

More details on these sorts of things can be found, e.g., on Robert Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics.