Does an Operator that neither commutes with $\hat{X}$ or $\hat{P}$, nor can be expressed as a "function" of $\hat{X}$ and $\hat{P}$ make sense?

Question

When you come from classical hamiltonian mechanics (which is based on the phase space), observables are introduced as functions $f$ on the phase space $(q, p)$. There can't be a classical observable that isn't a function of $q$ and $p$ by definition. In quantum mechanics, however, $\hat{X}$ and $\hat{P}$ are operators, acting on an infinite dimensional hilbert-space, so it seems at least imaginable to me that there are operators that can't be expressed as a "function" of $\hat{X}$ and $\hat{P}$.

I put the "function" into quotation marks because I don't know how to rigorously define such a function on the space of operators, put asside a taylor expansion using $\hat{X}$ and $\hat{P}$ as factors.

Of course I can just make the hilbert-space "bigger", for example by introducing a new spatial dimension $\hat{Y}$ with associated momentum $\hat{P}_y$, but these new observables would automatically commute with the former ones ($\hat{X}$ and $\hat{P}_x$) because they act on another subspace of the hilbert-space. I have a hard time imagining an operator which doesn't commute with ($\hat{X}$ and $\hat{P}_x$), while at the same time still not depending on each of them. Can somebody provide either an example for such an operator, or give a proof why such an operator can't exist?

EDIT: To clear 2 misconceptions that did arise:

• When talking about operators, I only ask about cases where the operators are linear
• By "function" I mean any operations that you can perform on operators (multiply them, add them, exponentiate them, have infinite sums or for the sake of the argument as well integrals). The question is about an operator that can't be expressed as "function" of $\hat{X}$ and $\hat{P}$ alone (that means I CAN'T FIND function in the above sense to relate these operators), but still does not commute with at least one of them.

EDIT 2: Since there is more than one answer now with (at first sight) contradicting content, it seems to me that the answers to the question do depend on further assumptions:

• Do we consider $X$ (or $P$) to be a complete set of operators, or equivalently, can I express any state as a linearcombination of $|x\rangle$, where the coefficients do depend solely on $x$.
• Do I require them to be self-adjoint, symmetric, and bijective?
• Am I restricting to "local" Operators?

Answer 1

In classical mechanics, any "observable" has to be a function of $x,p$ because we define $x,p$ to be the degrees of freedom of the system. If we experimentally find an observable that does not depend on $x,p$, it means there were extra degrees of freedom that we forgot to include. If so, we just enlarge our phase space, i.e., we include these extra degrees of freedom among the $x,p$. So in the end, any observable can always be written as a function of the phase-space variables.

In quantum mechanics, the philosophy is exactly the same. Here, "funcional dependence" is replaced by the notion of completeness: a set of operators $\{\mathcal O_i\}$ is said to be complete if $[A,\mathcal O_i]=0$ implies $A\propto 1$, the identity operator.

When specifying a classical system, one must declare what the coordinates are. When specifying a quantum system, one must declare what a complete set of operators is. A typical example is $X,P$, which is often assumed to be complete. Some problems require extra degrees of freedom, such as the intrinsic spin $S$, in which case a complete set of operators would be $X,P,S$. One may imagine systems that require more degrees of freedom, but also ones that require less (say, finite-dimensional systems).

Being "complete" is the formalisation of the notion of expressing an operator as a function of other operators. In particular, if we declare that $\{\mathcal O_i\}$ is complete, then any other operator $T$ can be expressed as a function of the $\mathcal O_i$, by definition.

Thus, if we declare that $X,P$ is complete, then any other operator must be expressible as a function thereof. In order to have an operator that cannot be expressed as such, one must assume that $X,P$ is not complete. So complete the set: $X,P,Q$, for some $Q$. Now we are free to define the $Q$'s to behave as we want. Say, the $Q$'s could be the position and momenta of extra dimensions, in which case there is a somewhat canonical (pun indented) notion of commutator, namely $[Q,X]=[Q,P]=0$. (But note that this is not forced upon us; it is perfectly consistent to assume that the extra dimensions do not commute with the old variables. For example, we could assume a curved configuration space, and so $[X_\mu,X_\nu]=\omega_{\mu\nu}$ for some form $\omega$).

But we could also take some $Q$'s that do not, by hypothesis, commute with $X,P$, in which case we would have $[Q,X]\neq 0\neq [Q,P]$. And we assumed that $X,P,Q$ is complete, but $X,P$ alone is not, which means that $Q\neq Q(X,P)$. So the answer to the question in the OP is: yes, you can definitely have an operator that does not commute with the canonical variables, yet is not expressible as a function thereof.

(At least, at a matter of principle it can exist: there is nothing inconsistent about the existence of such an operator. But experience tells us that the canonical variables are always enough to encode all the relevant degrees of freedom, and so they are always in practice complete. If we find a system where it is not, it means we did not realise there were some extra degrees of freedom, and we have to include those together with the canonical variables, i.e., they become canonical variables themselves.)

Answer 2

Let's say we're working in a Hilbert space of square-integrable functions $f(x)$. Indeed it's natural to talk about the operators $X$ and $P$ which act in a simple way on $f(x)$. In principle, you can talk about any linear operator that acts on $f$. For instance, you could try to solve the Schrödinger equation $$(P^2 + X^2 + T)\psi(x) = E \psi(x)$$ where $$T\psi(x) = \int dy\, Q(x,y) \psi(y)$$ and $Q(x,y)$ is your favorite function. If $Q(x,y)$ is proportional to a delta function, say $$Q(x,y) = \delta(x-y) V(x)$$ then this is nothing but the standard Schrödinger equation with a potential $V(x)$. But for a generic function $Q(x,y)$, you can't express the operator $T$ as a finite sum of $X$es and $P$s.

Often in physics, we rule out such Hamiltonians. The reason is that the interaction is not completely local: at a fixed time $t$, operators at points that are very far away (spacelike separated) interact. In a fundamental theory, say in particle physics, this type of interaction would violate causality. But in effective theories (say in condensed matter), such Hamiltonians can and do arise.

Answer 3

An operator can be defined by the comutators it has with respect to x and p. Suppose we have an operator $y$ that has a non-trivial commutation relation with $x$, i.e.

$$ [x,y]=f(x,p,y),\qquad [p,y]=g(x,p,y) $$

Note that ordering is determined by $f$ and $g$. We can always solve this equation by the ansartz $y(x,p)$, up to some integrability condition.

Answer 4

The total angular momentum, involved in the spin-orbit interaction: \begin{equation} {\bf J} = {\bf L} \oplus {\bf S} \end{equation}

Cannot be expressed as a function of $X$ and $P$ (because of the spin part ${\bf S}$), and it doesn't commute with them either (because ${\bf L}$ doesn't).

Answer 5

I think there does exist such a general expression for an arbitrary linear operator: $$ \hat O = \int dy \exp\left[i\hat p(\hat x-y)\right] \, O(y,\hat x), $$ where $y$ is a c-number and $O(y,x)$ is a function satisfying $O(y,x) = \langle y|\hat O|x \rangle$.

So the answer is no.

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Update: I realized my original answer is closely connected to the Wigner function: $$ O_W(x,p) \equiv \int dy \exp\left[ip(x- y)\right] \, \langle y|\hat O|x\rangle. $$ My original answer is equivalent to the following. For an arbitrary operator $\hat O$, compute its Wigner function $O_W(x,p)$, and we can express $\hat O$ in terms of $\hat x$ and $\hat p$ via $$ \hat O = O_W(\hat x, \hat p). $$ [$O_W(\hat x, \hat p)$ is nothing but the Weyl quantization of $O_W(x, p)$.] Indeed, it can be verified that the two operators on both sides have the same Wigner function, which is $O_W(x,p)$, so the two operators are equal.