Additional note on quant-ph/9506042

By T.Sakaguchi

In the paper, if we take Eq.(15) and ask what happens if a = b, we may be able to say that the observer state branches into the states

|[gamma_{live}]> + |[gamma_{dead}]>                                (C1)

and

|[gamma_{live}]> - |[gamma_{dead}]>,                               (C2)

in which the observer may see the superpositions of the states |gamma_{live}> and |gamma_{dead}>:

U(t)(|gamma_{live}> + |gamma_{dead}>)|[]>
  = |gamma_{live}>|[gamma_{live}]> + |gamma_{dead}>|[gamma_{dead}]>
  = 1/2 (|gamma_{live}> + |gamma_{dead}>)(|[gamma_{live}]> + |[gamma_{dead}]>)
  + 1/2 (|gamma_{live}> - |gamma_{dead}>)(|[gamma_{live}]> - |[gamma_{dead}]>).
                                                                   (C3)

Since this question seems to be common to all people who claim that Everett's formulation of QM is incomplete and advocate the importance of decoherence, let me explain how this can be resolved. The error comes from a misunderstanding of the states |[gamma_{live}]> +- |[gamma_{dead}]>.

As explained in the paper, in order for the observer to be able to see the states |gamma_{live}> +- |gamma_{dead}>, the corresponding observer states |[+-]> have to satisfy the following equation:

U(t)(|gamma_{live}> +- |gamma_{dead}>)|[]>
  = (|gamma_{live}> +- |gamma_{dead}>)|[+-]>,                      (C4)

However, comparing this equation with Eq.(C3), we can see,

|[+-]> != |[gamma_{live}]> +- |[gamma_{dead}]>.                    (C5)

That is, the states |[gamma_{live}]> +- |[gamma_{dead}]> do not correspond to the individual observer states |[+-]>, but simply show the superpositions of the observer states |[gamma_{live}]> and |[gamma_{dead}]>. The point is that the observer states are described by |[...]> and are independent of the places where the state vectors appeared in the equation. The measures assigned to the observer states |[gamma_{live}]> and |[gamma_{dead}]> in Eq.(C3) are invariant under any basis transformations (therefore, there is no need for a preferred basis).

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