# Why Superselection Rules are Physically Impossible

• Demystifier
In summary, Superselection rules are a set of phenomenological rules that are not derived from first principles but instead incorporated into theories as either axioms or theorems. These rules are observed in nature and can be explained by decoherence, which picks out certain bases in the Hilbert space as "preferred" ones. Further research and understanding is needed to fully explain the origin and implications of superselection rules.

#### Demystifier

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

Consider an example from another thread:
dextercioby said:
For example charge in case of a Dirac equation/field. The linear superposition between an eigenfunction with positive charge (positron) and negative charge (electron) makes no sense physically ...
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 allows 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 ?'

dextercioby said:
Am I getting this wrongly, or are you questioning whether a fundamental theoretical origin for the known superselection rules does exist ?
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).

dextercioby said:
Am I getting this wrongly, or are you questioning whether a fundamental theoretical origin for the known superselection rules does exist ?
You get this right.

dextercioby said:
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.
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.

dextercioby said:
To me 'Why are there superselection rules ?' is as valid a question as 'why is there a CPT invariance ?'
There is a CPT *theorem* derived either from axiomatic QFT or from concrete Lorentz invariant actions. Nothing like that seems to exist for SR.

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tom.stoer said:
Then the state |1> + |-3> is physically nonsense.
Why?

tom.stoer said:
What goes wrong physically or mathematically when I try to construct this state? What imposes the superselection rule?
Shouldn't those questions by answered by the person who claims that they are physically nonsense? (You in this case.)

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}
}

A. Neumaier
Demystifier said:
Why?

Shouldn't those questions by answered by the person who claims that they are physically nonsense? (You in this case.)
I tried to answer in post #4

Demystifier said:
[...]
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. [...]

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.

dextercioby said:
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.

Please tell me an example for a superselection rule which is realized in nature and how it could be derived or motivated.

DrDu said:
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}
}

tom.stoer said:
Please tell me an example for a superselection rule which is realized in nature and how it could be derived or motivated.

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.

DrDu said:
... the impossibility to superpose a fermionic and a bosonic state?
Great example.

Any idea why this is forbidden? Parity?

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? :-)

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Thanks, sounds reasonable.

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.

strangerep said:
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? :-)

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.

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

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.
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 $J^2$ 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 $J^2 + K\cdot P + P\cdot K - 2D^2$ (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 transitions 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.

strangerep said:
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...

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 $\langle 2| A | 1\rangle=0$, whether A is the generator of a symmetry or not.

DrDu said:
E.g. for optical transitions 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.
I see the following problem with it. The total angular momentum in a transition must be conserved. So, if the atom changes its angular momentum, then the opposite amount of angular momentum must be taken away by another particle, say photon. But then the total pure state is an entanglement between atom and photon. Consequently, the atom itself is not in a pure coherent state. In other words, we cannot longer say that atom is in a coherent superposition.

Demystifier said:
I see the following problem with it. The total angular momentum in a transition must be conserved. So, if the atom changes its angular momentum, then the opposite amount of angular momentum must be taken away by another particle, say photon. But then the total pure state is an entanglement between atom and photon. Consequently, the atom itself is not in a pure coherent state. In other words, we cannot longer say that atom is in a coherent superposition.
The robust eigenstates of a photon field are not the ones with fixed photon number but coherent states with unsharp photon number. Upon interaction with the atomin its ground state, the coherent state of the photon field will not change and the wavefunction be separable.

Probably the argument would be simpler if considering directly photon fields. The spin of a photon is not well defined but it's helicity is. Like with spin, states of different absolute value of helicity correspond to inequivalent representations of the Poincare group. Nevertheless, coherent photon states (with unsharp absolute value of helicity) exist and their phase is observable.

DrDu said:
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.
Yes, but they too have a larger dynamical group.

E.g. for optical transitions 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.

In such cases, we should think in terms of the full Hamiltonian of (say) the atom in interaction with a radiation field. The group action generated by the full Hamiltonian (i.e., time evolution) constitutes a continuous (physical) transformation from the one spin state to another -- though of course one must include the photon or radiation field to get the full picture.

Probably the argument would be simpler if considering directly photon fields. The spin of a photon is not well defined but it's helicity is. Like with spin, states of different absolute value of helicity correspond to inequivalent representations of the Poincare group. Nevertheless, coherent photon states (with unsharp absolute value of helicity) exist and their phase is observable.

An ordinary coherent state is an eigenstate of the annihilation operator, with eigenvalue $\alpha$, say, which corresponds to the complex amplitude of the classical EM wave mode. This complex amplitude can be varied continuously -- we can adjust the amplitude and phase of the wave. Similarly for multimode coherent states, a continuous symplectic/Heisenberg group still determines the space of all such states (via the action of the group on a vacuum state). Hence this example still does not contradict the proposition that a superselection rule only applies in the absence of a continuous physically-meaningful group of transformations.

That's why I was asking about your group concept. If you consider also quite general dynamical groups, i.e. those groups which transform any eigenstate of a hamiltonian into any other one, then you are probably right. However, I wonder whether this would be generally be considered as a "symmetry" group.

DrDu said:
[...] If you consider also quite general dynamical groups, i.e. those groups which transform any eigenstate of a hamiltonian into any other one, then you are probably right. However, I wonder whether this would be generally be considered as a "symmetry" group.

OK, yes, the term "symmetry" group is distinct from "dynamical" group and my earlier post should be modified accordingly...

Cheers.