Correction to: On reconstructing parts of quantum theory from two related maximal conceptual variables. (Intern. J. Theor. Phys. 61, 69, 2022.)

Correction to: International Journal of Theoretical Physics (2022) 61: 69

https://doi.org/10.1007/s10773-022-05047-4

A serious misformulation in my article is corrected.

The main result of this article, Theorem 1 of Section VI, point (ii) (page 7 of 19), should read:

Theorem 1 Consider a situation where there are two maximal accessible conceptual variables θ and ξ in the mind of an actor or in the joint minds of a communicating group of actors. Make the following assumptions:

• (i) On one of these variables, θ, there can be defined transitive group actions G with a trivial isotropy group and with a left-invariant measure ρ on the space Ω_θ.

• (ii) There exists a unitary multi-dimensional representation U(·) of the group behind the group actions G defined on θ such that the coherent states $U(\mathrm{g}){\left|\theta \right.}_{0}\rangle$ are in one-to-one correspondence with the values of g and hence with the values of θ.

• (iii) The two maximal accessible variables θ and ξ can both be seen as functions of an underlying inaccessible variable $\phi \in {\Omega }_{\phi }$. There is a transformation k acting on ${\Omega }_{\phi }$ such that ξ (ϕ) = θ(kϕ).

Then there exists a Hilbert space ℋ connected to the situation, and to every accessible conceptual variable there can be associated a symmetric operator on ℋ.

The important change is that in point (ii) the representation U(·) is not assumed to be irreducible. It is crucial that this is not assumed. This is the whole point of the proof on the pages 9-12 (see for instance Lemma 1 on page 10). In this proof, the representation may be reducible or irreducible. The important point is that it is multi-dimensional, so that the given coherent states can be in one-to-one correspondence with the group elements.

For the applications of Theorem 1 is is crucial that reducibility is permitted. When the representation U(·) is not assumed to be irreducible, it is possible to show that in very many cases the assumption (ii) can be automatically fullfilled. Consider here the case where the maximal variable θ takes n different values, so that the Hilbert space ℋ has n dimensions:

In this case, let G be the cyclic group acting on the distinct values u₁,…,u_n of θ, that is, the group generated by the element g₀ such that g₀u_i = u_i+1 for i = 1,…, n−1 and g₀u_n = u₁.

This is an Abelian group, which only has one-dimensional irreducible representations. However, we can define U(·) as taking values as diagonal unitary n × n matrices with different complex nth roots of the identity on the diagonal. For the specific matrix U(g₀), take these nth roots in their natural order, and then let every element of G be mapped into the diagonal matrices U(·) by the corresponding cyclical permutation.

It is easy to see then that the coherent states $U(\mathrm{g}){\left|\theta \right.}_{0}\rangle$ are in one-to-one correspondence with the group elements g $\in$ G when ${\left|\theta \right.}_{0}\rangle$ is a unit vector with one element equal to 1 and the others zero. Also, G is transitive on its range and has a trivial isotropy group. Thus the only assumption of Theorem 1 that is left to verify, is the assumption that ξ can be found as a related variable to θ, that is, the existence of an inaccessible variable ϕ and a transformation k in the corresponding space ${\Omega }_{\phi }$ such that ξ (ϕ) = θ(kϕ).

In the special case where θ = θ^a and ξ = θ^b are two different components of a spin/angular momentum vector, the last requirement can be satisfied by letting ϕ be the projection of the full spin/angular momentum vector upon the plane spanned by the two components, and taking k to be the 180° rotation around the midline between the two component directions.

In this special case, we can let the group K be the group of rotations in the above plane. Then θ(·) is permissible with respect to this group, where the notion of permissibility is defined in the article. (Note that k does not belong to K in this case, but this is not necessary.)

Further developments, generalizing this special case, are under consideration now.

Also, this Theorem 1 should have the following addition: The case ϕ = (θ,ξ) with k just exchanging θ and ξ, should be explicitly excluded. The proof of Lemma 1 does not work for this case. If this case had been allowed, all maximal accessible variables would then by definition have been related.

Finally, I take this opportunity to correct two misprints on page 10 in the proof of Theorem 1:

On line 7, replace g jψ(ϕ) = (gθ(kϕ), θ(ϕ)) by g jψ(ϕ) = (gθ(kϕ),gθ(ϕ)).

On line 8, replace h jψ(ϕ) = (ξ(ϕ),hξ(k⁻¹ϕ)) by h jψ(ϕ) = (hξ(ϕ),hξ(k⁻¹ϕ)).