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In his Quantum Mechanics: A Modern Development, Ballentine writes (page 184, second edition) that "It is sometimes asserted that states that would be described by vectors [which are linear combinations of eigenstates from different eigenspaces of operators defining superselection rules] do not exist." He then goes on to say that this statement is not strictly speaking true (unless one assumes it), and in fact that all we can say is that the relative phase between elements of different eigenspaces of an operator defining a superselection rule can have no observable consequences.

On the other hand, in his Symmetry Principles in Quantum Physics, Fonda says (bottom page 10) that linear combinations of of eigenstates from different eigenspaces of operators defining superselection rules simply do not exist. He says, for example, that no experimenter has ever observed a state which is a superposition of states with different charges.

What is the consensus here? Does it matter that Ballentine was talking about superselection given by rotation by $2\pi$, whereas Fonda was talking about charge? I wouldn't imagine that certain superselection operators are privileged above others?

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  • $\begingroup$ Perhaps this boils down to what both authors mean with state (as opposed to vector)? $\endgroup$
    – Tobias Fünke
    Commented Aug 5, 2023 at 17:27

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There's nothing stopping you from simply writing down a sum of two state vectors in different superselection sectors. However, these sums are completely incoherent. That is, the expectation values and probabilities prescribed by the new state are identical to those of the mixture. Thus, they are the same state. Because the physical properties that we associate with superpositions are really properties of coherent superpositions in which there exist operators mixing the two sectors, it is common parlance to say that a superposition between two sectors does not exist. Only superpositions of states within the same sector are superpositions in the traditional sense.

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  • $\begingroup$ I am not quite sure I follow. Are pointing out that, though two different states are necessarily represented by different rays, the converse need not hold. That is, two different rays may represent the very same state (and, presumably, such a coincidence occurs if and only if said vectors involve superpositions of vectors from different incoherent subspaces). Thus there is no difference (if I consider superselection by rotation by $2\pi$) between... $\endgroup$
    – EE18
    Commented Aug 5, 2023 at 20:04
  • $\begingroup$ ...$|\psi_+ \rangle = |+\rangle + |-\rangle$ and $|\psi_- \rangle = |+\rangle - |-\rangle$ (or indeed any |\psi_\omega \rangle = |+\rangle + e^{i\omega}|-\rangle). But how does this imply that I can somehow ignore states which are written as such as linear combinations of different incoherent subspaces? Is the idea that they behave as (impure) states and so I can say the system is in fact being drawn from a system represented by something like $\rho = C(|+\rangle\langle +| + |-\rangle\langle -|)$ normalized by $C$ as required? And then this impure state leads me to... $\endgroup$
    – EE18
    Commented Aug 5, 2023 at 20:06
  • $\begingroup$ ... the choice of interpretation as an incoherent mixture? $\endgroup$
    – EE18
    Commented Aug 5, 2023 at 20:07
  • $\begingroup$ Oh sorry I should've clarified. In more complicated theories that have superselection rules, a state is defined not as a vector, but as an expectation functional. If two vectors or density matrices have the same expectation values for every physical observable, then they are two representations of the same state. $\endgroup$
    – Prox
    Commented Aug 5, 2023 at 20:34
  • $\begingroup$ I see. Is my statement fair under the nonrelativistic QM rubric? That is, we can say or choose that there are no linear combinations of elements in different incoherent eigenspaces since, for example, the state $|\psi_+ \rangle = |+\rangle + |-\rangle$ (in operator form, $\rho_{\psi_+} = |+\rangle\langle +| + |+\rangle\langle -| + |-\rangle\langle +| + |-\rangle\langle -|$ gives the exact same predictions (and therefore represents the exact same state) as $\rho = |+\rangle\langle +| + |-\rangle\langle -|$ (and this latter state can be thought of as an incoherent/impure mixture)? $\endgroup$
    – EE18
    Commented Aug 5, 2023 at 20:50

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