## 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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© 1996-2017 Toshifumi Sakaguchi