Why we need superselection rules

One motivation for superselection rules comes from a passage in von Neumann's famous Mathematical Foundations of Quantum Mechanics:

There corresponds to each physical quantity of a quantum mechanical system, a unique hypermaximal Hermitian operator, as we know..., and it is convenient to assume that this correspondence is one-to-one -- that is, that actually each hypermaximal operator corresponds to a physical quantity. (von Neumann 1932 [1955], p.313)

The problem is that von Neumann's "convenient assumption," that self-adjoint operators correspond to observable quantities, can't possibly be true. Here's a (probably too)* simple example to illustrate. Suppose we have a 2-dimensional Hilbert space (it might be describe the z-spin of a single particle), generated by the states ψ₀ and ψ₁. And let A be a self-adjoint operator for which these are eigenstates:

Construct two new states, φ₀ and φ₁, given by:

Now let B be a self-adjoint operator for which these new states are eigenstates, say,

The problem is that, although both A and B are self-adjoint, they can't both at the same time represent observable quantities. For suppose the eigenstates of A correspond to observable measurements; then the eigenstates of B are not observable states, since they are linear superpositions of the eigenstates of A. But it's not possible to observe superpositions of physics measurements. And the same holds conversely if the eigenstates of B correspond to observable measurements. So, which one is the real observable? It seems we're in a pickle.

One simple way to resolve the pickle is to impose a rule, which says constructed operators like B are not allowed to be observables. That's a superselection rule.

A typical way to implement such a rule is to notice that we can think of our Hilbert space as a direct sum of one-dimensional Hilbert spaces H₀ and H₁, one containing ψ₀ and the other containing ψ₁:

We can motivate this decomposition as long as we agree that both ψ₀ and ψ₁ correspond to readings of a measurement device. We can then impose a superselection rule:

Linear superpositions of states from distinct sectors in such a direct sum are not physically realizable except as a mixture, and hence cannot be eigenstates.

Wightman attributes this particular rule to Wigner, in his excellent history of superselection rules. John Earman has recently written on some of the philosophical details. Of course, it may not be true that "we agree" on what states correspond to readings of measurement devices. And in such cases, the question of which superselection rules are justified remains a difficult problem in the foundations of physics.

* Added.