Does the self-adjointness of an operator in quantum mechanics "depend" on the representation?

Question

This question mainly arises from thoughts concerning the quantum system of a single particle in the real half-line.

As it is known from the literature, a careful analysis of the position and momentum operator for this system in the position representation, where the operator $\hat{x}$ acts as a multiplicative operator and the operator $\hat{p}$ acts as a differential operator, each in a proper domain, leads to conclude that the momentum operator $\hat{p}$ is not essentially self-adjoint and it does not admit any self-adjoint extensions. In other words, the momentum, defined as above, cannot be an observable for the system.

(I summarized this part very briefly, assuming all the details are known, but to be clear the analysis I am referring to is analogue to the one discussed here: What's the deal with momentum in the infinite square well?)

This also should suggest that the p-representation is not a physical representation and hence it is not viable.

Nevertheless, if I start again from scratch, assuming that the only content of my knowledge is the algebra $[\hat{x},\hat{p}]=i\hbar$ and the configuration space represented from the positive real half-line, in principle I could make use of the p-representation. But in this representation, where the operator $\hat{p}$ acts just as a multiplicative operator in a proper domain, I cannot see how I can obtain the same conclusion about its self-adjointness. And if this is true, that is that in this representation p can be self-adjoint, how it is possible to conciliate this result with the former one about the observability of momentum?

I can see that I am handling two different operators in different spaces and from this perspective, it could be reasonable to obtain that one does not admit self-adjoint extensions while the other does. But from a physical point of view concerning the status of observable of a theory I am lost.

So, there should be a loophole somewhere for sure, but I am not able to find it.

Thanks all in advance.

Answer 1

By the standard axioms of the textbook Quantum Mechanics, momentum in a direction, coordinate, spin, and Hamiltonian are fundamental observables, which means that in whatever physical system we should try to realize them as self-adjoint operators, ideally all. In systems where coordinate and momentum along the coordinate are "conjugated" observables, one is free to choose a Hilbert space (since all of them are isomorphic to one another) as $L^2$.

We thus have: $$ \displaystyle{\left\{x, p_x, L^2 \left(\Omega\subseteq\mathbb{R}, dx\right)\right\} \longleftrightarrow\left\{\tilde{x}, \tilde{p_x}, L^2 \left(\tilde{\Omega}\subseteq\mathbb{R}, dp\right)\right\}} $$

So in the coordinate representation, $x$ could easily be self-adjoint, but $p_x$ could only be symmetric (with no self-adjoint extensions). Change the Hilbert space by a Fourier transformation and "in momentum space" $p_x$ could easily be rendered self-adjoint, but the $x$ would be again symmetric (with no self-adjoint extensions).

So the correct steps for any problem are:

• describe the system (for example a spinless particle in a certain potential).
• according to the potential, determine the concrete form of the Hamiltonian as a function of the other observables.
• after the Hamiltonian is determined, you need to choose the Hilbert space appropriate to solve the spectral problem for the Hamiltonian. This usually means playing with a differential equation and boundary conditions.
• with the chosen Hilbert space, ensure that the Hamiltonian is self-adjoint, i.e. the "physical boundary conditions" in the ODE make the Hamiltonian self-adjoint as a differential operator. See if the self-adjointness of the Hamiltonian automatically entails the self-adjointness of the other fundamental observables (coordinate, momentum, angular momentum, spin). If not all, tough luck.