When superposition isn't allowed, that is called a superselection rule.

Derek, I am not quite decided whether I'd accept your question as being posed in an appropriate manner. In QM what you can superimpose is not "variables" but "states". Variables (observables) have expectation values in states. In a quantum superposition of states these expectation values are those or the superimposed states.

So I would go for the question as to what are the observables. I'd say it depends o how you construct your QM framework. In "standard" QM we start from a Lagrangian, calculate conjugate variables and proceed to quantization by introducing corresponding operators (in whatever representation appears appropriate).

There is a mapping from classical variables to their quantum counterparts (with the exception of those variables/quantities that do not exist classically, such as e.g. the finite angular momentum (spin) of a point charge (electron)). Every classical vaiable O within the theoretical framework we set up in the Lagrangian can be described as a function of the canonical variables. The operator for the quantum observable is obtained by the same functional prescription, now applied to the operators representing the canonical variables (while respecting the commutators appropriately).

So my suspicion is the following: any quantitiy that you formulate classically as given (such as e.g. the electron charge in a typical motion-in-potential type problem) then it will also be given in the QM formulation. It then does not make sense to ponder a superposition. It might look different if you were to set up a theory which introduces other (field) quantities with the aim to describe charge quantisation. In so far as charge is then a property of states which could be superimposed, then so will charge be (in the sense of the expectation value of the charge operator, whatever it may look like). I may be wrong here, but this is how I think about it.

The same question applied to the classical variables pressure, temperature, entropy etc. maybe has a different flavor, since the most intuitive approach (for me) to envisage them is via ensembles of (ideally or approximately noninteracting) quantum objects. The ensemble variables then do not relate to the individual states following from QM (solution of Schrödinger equation) but represent the ensemble. Therefore the QM superposition in the sense above is meaningless in this case.

Superpositions will always be superpositions of states, i.e. the "wave function" of the "whole thing". Pressure would relate to the derivative of total energy with respect to volume, so it can be related to the many body quantum state. Entropy is rather a property of the whole set of sulutions (essentially the log of the density of states). Temperature then can be introduced as the energy derivative of entropy.

As explained here http://physics.stackexchange.com/questions/56545/what-really-are-superselection-sectors-and-what-are-they-used-for it is not the case that superpositions of states with different charges etc. are disallowed, it's just that we can never experimentally show that a system exists in such a state: since the charge is exactly conserved no experiment can be arranged to interfere states with different charges. The superselection rule business is just a convenience: since superpositions across superselection sector have no experimental import you can restrict attention to states which lie in a single sector.

http://physics.stackexchange.com/questions/56545/what-really-are-superselection-sectors-and-what-are-they-used-for

Quantization doesn't enter into that question. It is but a method to get a quantum theory from an old known classical one, but quantum mechanics stands by itself. We start from a Hilbert space and define linear operators thereon, including the Hamiltonian. So we get charge, strangeness, charm, color, isospin, and many other quantum numbers that have no classical counterpart. Most of the time they are linked to the representations of a symmetry group, just like spin, and that is also true for momentum, energy etc. States of different charge, strangeness, charm, color, isospin can be superposed, since they are defined quantally.

Claude, I have no objection to your statements. And we seem to agree on states being superimposed, not variables. (In what I wrote above, I intended to make a connection between variables & observables and to contrast them to states).

Just as a side question: If superposition is superposition of states, it seems to me that we are thinking in terms of the Schrödinger representation. Is it possible to describe superposition also in the Heisenberg representation? It should be possible, but I never have seen it mentioned, let alone the gory details...

@Herbert: Whenever the Hamiltonian leaves coherent subspaces invariant (I know of no exception to this) then the superselection operators (i.e. these self-ajoint operators that commute with all observables) are related in Heisenberg- and in Schroedinger picture in just the same way as all normal observables. No gory details!

So going back to the spirit of my original question: can we have a superposition of mass? If not, why not?

Derek: For mass we can, neutral Kaons are an example. Neutrino oscillations (still hypothetical?) are another example.

So can we make an agreed list of those which can exist and those which cannot?

Quantum superposition applies to The Hilbert space state vector of a system. In the special case where the system is describable by a wave function, the quantum superposition principle applies to the wave function. If Probability amplitudes are being calculated by a Feynman path integral method, then the amplitudes for different paths must be added linearly, another illustration of quantum superposition.

I think the question is related to the so-called "Super-Selection Rule". (Chapter 2 Sec. 4 of Martin and Spearman's book "Elementary Particle Physics")

According to SSR, one can not prepare a superposition state of certain conserved quantum variables. According to my understanding of SSR, these quantum variables is 1 exactly conserved (just as the textbook says), 2 the corresponding operator commutes with all the other conserved operators, 3 the operator can only take discrete values. 2 and 3 is just my intuitive understanding.

SSR tells us that not only the corresponding operators is not only exactly conserved but also that all the "physical states" are the eigenstates of the corresponding operators.

By physical state, I think it means state that we can prepare. Mathematically, one can write down a superposition state, but we can not create this state by physical method.

An interesting example is the superconductor. The microscopic Hamiltonian conserves electron number, whereas all the experimental evidence is consitent with the BCS theory, in which the effective Hamiltonian (obtained after symmetry breaking) does not conserve the electron number. So, sometimes, even if you do not have a term in the Hamiltonian that allows superpositions, the system may behave as if there was one, and to an astonishing precision! I just wanted to point out that the super-selection rule can be lost in emergent phenomena.

@Vitaly, your essay is interesting but you seem to mix things. As long as the Hamiltonian is a proper linear operator (it is so nearly by definition, I know of no quantum system that gives rise to a non-linear Hamiltonian) it can't forbit that superpositions of states can be formed. At least in the history of superselection rules these limitations came from other players, e.g. the charge operator.

For fermions particle number and charge are nearly synonymous. Superpositions of 0 and 1 electrons in vacuum are forbidden. But superpositions of 0 and 1 cooper pairs (which is roughly like 0 and 2 electrons) are allowed in a cooper-pair box, and are routinely created. Can you explain what I mix up, please? Just want to understand this question once and forever :).

Vitaly,

I am not familiar with BCS theory, so correct me if I'm wrong.

Is the superposition of 0 and 1 cooper pair actually an entangled state:

|0 cooper pair>|the rest of the system, charge 0>+|1 cooper pair>|the rest of the system, charge +2>?

In the effective (BCS) description, the fermionic number operator is no longer conserved, due to symmetry breaking. With such a Hamiltonian one can create superpositions of different even number of fermions, since one has at hand a term like psi^+psi^+ plus its hermitian conjugate psi psi. In a cooper-pair box true superpositions are created, suitable even for quantum computing to some extent. Later more clever qubits were invented.

From the microscopic point of view (in terms of bare particles) the state is much more complicated than you describe. It is similar in nature to the case when you shine a laser beam onto an atom and create |0>|laser> + |1>|laser>. The laser state hardly changes if you take out a photon. The BCS state is similar, but it is of the fermionic type. Both go under the name of coherent states, i.e. eigenstates of a destruction operator.

My point was that super-selection rule is not fundamental. People just found a system (called this world) where there were no terms in the "Hamiltonian" (no interactions in nature) that would allow to create superpositions. If you have no means of creating superpositions, then you have to admit that they cannot be created. Maybe tomorrow new interactions will be discovered and that will no longer be the case. But other than that, the super-selection rule is like any other symmetry.

@Vitaly,

now it looks to me that the person who mixed things up was I. Unfortunately so far I did not reach your ideal ' to understand this question once and forever', but plausibility is more with you than with my original view. Thanks for your inspiring thoughts.