A superselection rule is a rule that forbids superposition of quantum states. As stated by Lubos here, one cannot superpose states with different charges because of the conservation of charge:
An example in the initial comments involved the decomposition of the Hilbert space to superselection sectors ${\mathcal H}_Q$ corresponding to states with different electric charges $Q$. They don't talk to each other. A state with $Q=-7e$ may evolve to states with $Q=-7e$ only. In general, these conservation laws must be generalized to a broader concept, "superselection rules". Each superselection rule may decompose the Hilbert space into finer sectors.
Alternatively, one might say we don't have to consider such superpositions; it doesn't matter if there are other branches of the wavefunction with different $Q$ because they'll never interfere, so we might as well toss them out. This doesn't affect the results of experiments.
I'm confused why this same logic doesn't forbid almost any superposition whatsoever. For example, we often talk about a superposition of spin states $$|\psi \rangle = \frac{|\uparrow \rangle + |\downarrow \rangle}{\sqrt{2}}$$ or a superposition of momentum states $$|\psi \rangle = \frac{|p = p_0\rangle + |p = - p_0\rangle}{\sqrt{2}}$$ despite the conservation of angular momentum and momentum. Why exactly does superselection not forbid these kinds of superpositions as well?
Superselection has been discussed on this site a few times, but I haven't been able to find an argument that applies to conservation of charge but doesn't apply to conservation of momentum; this isn't a duplicate! One thought I had was that when we prepare a superposition of momentum states, we aren't really breaking superselection because there is a backreaction on the preparation apparatus, so we really have $$|\psi \rangle = \frac{|p = p_0, \text{app. recoils back}\rangle + |p = - p_0, \text{app. recoils forward} \rangle}{\sqrt{2}}$$ and the two states do have the same momentum. Then the original state we proposed is attained by just tracing out the apparatus; this doesn't decohere the superposition as explained here. This sounds plausible to me but then I don't understand why the same couldn't be said for charge conservation, leaving us with no superselection rules at all. (It's true that charge is discrete, but there are supposed to be superselection rules for continuous conserved quantities too.)
Answer
Superselection makes sense not abstractly in an arbitrary Hilbert space but only in Hilbert spaces structured by the introduction of an algebra of distinguished observables of interest (i.e., that can be combined to prepare states), with prescribed commutation rules. These define the physics that is possible in the class of models considered, and superselection is a concept defined relative to these. (In particular, if the class of relevant observables changes, the concept of superselection changes with it. Enlarging the observable algebra may merge some sectors but typically creates others.)
Typical examples are the universal enveloping algebras of Lie algebras, or the $C^*$ algebras generated by the corresponding Lie groups. For example Heisenberg algebras and Heisenberg (or Weyl) groups correspond to canonical commutation rules, which are the basis of much of quantum physics.
The Hilbert spaces of interest are the (continuous unitary) irreducible representation spaces of these algebras, Lie algebras, or groups. These (or more precisely the classes of equivalent such representation spaces) are called the (superselection) sectors of the theory. Since they constitute different Hilbert spaces it makes no sense to superimpose vectors of the different sectors. One can define an inner product on the direct sum of these Hilbert spaces, but the algebra of operators still map each sector into itself, hence there is no way to create (in a physically relevant way) a superposition from pure states within the sectors.
For finite-dimensional Heisenberg algebras/groups, all continuous unitary irreducible representations are equivalent (Stone-von Neumann theorem); hence for nonrelativistic N-particle theories, there are no superselection rules (that would specify superselection sectors).
Once one also accounts for spin, the situation becomes more complicated: a mixture of a fermionic and a bosonic state makes no longer physical sense since the two state vectors behave differently under a rotation by 360 degree - though formally it is still defined. No amount of new physics will change that.
For infinite-dimensional Heisenberg algebras/groups as they occur in (relativistic or nonrelativistic) quantum field theory, the Stone-von Neumann theorem is no longer valid, and there are uncountably many inequivalent continuous unitary irreducible representations, hence there are uncountably many superselection sectors, distinguished by their essentially different behavior at spacelike infinity.
In more technical terms: The most interesting superselection rules, accounting for superconductivity, charge, baryon number, etc., arise due to unimplementable Bogoliubov transformations, involving limits that are so singular that they lead out of the Hilbert space representing the vacuum sector. In particular, charged states have a sufficiently different asymptotic structure from uncharged ones, since the Coulomb field is long range, and they belong to different superselection sectors. This is a general property of charges in gauge theories, see Strocchi, F., & Wightman, A. S. (1974). Proof of the charge superselection rule in local relativistic quantum field theory. Journal of Mathematical Physics, 15(12), 2198-2224. No amount of new physics will change that.
Under certain conditions, superselection sectors can be classified; see, e.g., the article DHR superselection theory from nLab. DHR stands for Doplicher, Haag, and Roberts; see, e.g.,
- Doplicher, S., & Roberts, J. E. (1990). Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Communications in Mathematical Physics, 131(1), 51-107.
Superselection rules have nothing to do with conservation laws. In spite of momentum conservation, states of different momentum can be superimposed since the Lorentz transformations that turn one momentum state into another are unitary and hence are defined on the same representation space.
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