Superselection rules claim that some superpositions are physically impossible.
But why are they so?

Consider an example from another thread:

But why such a superposition makes no sense physically?

Clearly, if such a state has been produced from a charge eigenstate, then such a superposition would violate charge conservation. Indeed, charge is conserved in the Standard Model of elementary particles. But still, it is not difficult to write down a field theory action in which charge is not conserved, and in which such superpositions can be created from a charge eigenstate.

Moreover, I don't see why such a superposition could not be an INITIAL physical state even in the Standard Model.

I believe the only true reason why certain superpositions never occur in nature is the fact that they are not stable under decoherence. Decoherence picks out some bases in the Hilbert space as "preferred" ones, while others, such as Schrodinger cat states, are not seen because (due to decoherence) they live too short to be observed.

But I am open also for different opinions and facts I am not aware of, so please share them here.

Simple question: consider the Fock space of QED (fermionic sector) with its superselection sectors according to electric charge, i.e. Q|> = q|> with q...,-2,-1,0,1,2,... Suppose we have

Q|1> = 1|1> where |1> represents 7 electrons and 8 positrons and
Q|-3> = -3|-3> where |-3> represents 15 electrons and 12 positrons.

Then the state |1> + |-3> is physically nonsense. In addition it is not a Q-eigenstate.

What goes wrong physically or mathematically when I try to construct this state? What imposes the superselection rule? (besides the fact that usually Gauss law forces total charge to vanish and allowes only |0> in the physical Hilbert space).

Am I getting this wrongly, or are you questioning whether a fundamental theoretical origin for the known superselection rules does exist ? I don't know the full answer, I suspect it to be <no>, yet I invite you to read pages 85,86,87 of 1st volume of Galindo & Pascual textbook. Examples for univalence and mass are discussed later in the book. AFAIK, a proper description is also built in the algebraic formulation, but the issue of a deep theoretical explanation still remains.

To me 'Why are there superselection rules ?' is as valid a question as 'why is there a CPT invariance ?'

Hm, to be honest, I never thought about this question in detail. The reason is the following:
1) looking at the momentum P (or some other constants of motion) instead of a charge Q we do not expect a superselction rule
2) looking at certain charges not related to local symmetries we do not expect superselction rules, either (flavor and neutrino mixing ...)
3) looking at charges related to local gauge symmetries I would say that they have to be zero in the physical sector! The physical Hilbert space is a charge singulet = charge-neutral subsector of the full Hilbert space. As a simple example take compact 3-space i.e. a three torus and integrate the Gauss law constraint in QED; this results in Q|phys> = 0

So I never thought about the simple example before (a state with 3 electrons and 5 positrons is ruled out by an exact cosntraint Q|phys>=0, not by a superselection rule).

If I understood them correctly, they say that SR (Superselection Rules) are merely phenomenological rules, rather than laws which can be axiomatized or derived from first principles.

There is a CPT *theorem* derived either from axiomatic QFT or from concrete Lorentz invariant actions. Nothing like that seems to exist for SR.

There's a very nice article by Wightman, one of the fathers of that concept:

@article{wightman1995superselection,
title={Superselection rules; old and new},
author={Wightman, AS},
journal={Il Nuovo Cimento B (1971-1996)},
volume={110},
number={5},
pages={751--769},
year={1995},
publisher={Springer}
}

Yes, but generally these <phenomenological rules> should always be incorporated in a theory, either as axioms, or as theorems resulting from a set of axioms. This should be valid for SR as well. Can't prove them ? Axiomatize them.

Well, it's not always a good approach. For example, one could say the same for the wave-function-collapse "axiom", but we know very well how many interpretational problems the collapse "axiom" causes.

On the other hand, both collapse rule and superselection rules seem, at least partially, to be explainable by decoherence.

How about univalence superselection, i.e. the impossibility to superpose a fermionic and a bosonic state?
Any rotation by 360 deg. would multiply the fermionic part by -1 leaving the bosonic part of the superposition unchanged. Hence the phase can not be observable.
In QFT's with an infinite number of degrees of freedom there is an infinity of sectors of which are separated by a superselection rule, e.g. the sector describing non-interacting particles and a sector with a non-zero coupling.

ISTM, it all boils down to which symmetries are truly physical, and the structure of their projective unireps.

If we think of states as rays, not vectors, then the state r3 which is a superposition of two states r1 and r2 corresponds to a rotation in the plane defined by r1 and r2. I.e., r3 lies in this plane. Thus, a superposition is only possible if a corresponding continuous physical symmetry exists that rotates rays in this plane.

The case of positive and negative charges is usually represented via a complex conjugation of some sort, and there's no continuous transformation that takes a quantity into its conjugate. A rep and its conjugate are inequivalent: they're not related by a similarity transformation.

Likewise for univalence -- there's no continuous physical symmetry that takes a boson into a fermion. This is a consequence of the inequivalence of reps corresponding to different values of total spin.

Or can anyone think of examples to the contrary? :-)

So superselection sectors are subspaces in Hilbert space for which no continuous symmetry exists which rotates one ray in one subspace to another ray in a different subspace.

As long as you don't define what a "physical" symmetry shall be, it will be hard to argue against it.

Concerning the univalence rule, this is not a consequence of the inequivalence of different reps for different values of the total spin. Especially there is no superselection rule for systems with different total spin in general. E.g. a superposition of an hydrogen atom with different values of J can both be experimentally created and detected.

I had in mind transformations that are physically implementable (although perhaps you'll complain that this is also not a definition). E.g., a rotational symmetry is physically implementable, since I can physically rotate an object system in the lab relative to its surroundings...

Yeah, I was wondering about that. Thanks for mentioning it.

The dynamical group for the Hydrogen atom is the conformal group (or its covering group SO(4,2)), which is larger than mere SO(3). The usual [itex]J^2[/itex] Casimir of SO(3) is no longer a Casimir of SO(4,2) afaik, although there's another quadratic Casimir which I vaguely recall is of the form [itex] J^2 + K\cdot P + P\cdot K - 2D^2[/itex] (when expressed in terms of the usual conformal generators)?

I don't recall the form of the other two Casimirs of the conformal group, though. (It has 3 Casimirs, right?)

Anyway, the point I'm wondering about is that since we're dealing with very different Casimirs for the H atom, the structure of the unireps is likely to be significantly different also.

I wasn't thinking in some special group properties of the hydrogen atom. You could take any other compound system like a heavier atom or a nucleus if you like.
E.g. for optical transistions there is a selection rule that |J1-J2| <=k<=J1+J2, where k is the multipole moment of the optical radiation (dipole, k=1, in the long wavelength limit). Hence with electronic dipole radiation, you can create and detect superpositions of different total spin.

Also the eigenstates of the field operator (which is an observable in that case) for massive chargeless vector bosons (e.g. rho or omega mesons) aren't eigenstates of particle number and hence also not of spin.

So the generator Q of the symmetry should be an observable?
Well that's more or less the definition of a superselection rule. If two states 1 and 2 are separated by a superselection rule, then any matrix element for an observable A [itex]\langle 2| A | 1\rangle=0 [/itex], whether A is the generator of a symmetry or not.