> On 8 Aug 2019, at 02:23, Bruce Kellett <[email protected]> wrote:
>
> On Wed, Aug 7, 2019 at 11:30 PM Bruno Marchal <[email protected]
> <mailto:[email protected]>> wrote:
> On 7 Aug 2019, at 14:41, Bruce Kellett <[email protected]
> <mailto:[email protected]>> wrote:
>>
>> Superpositions are fine. It is just that they do not consist of "parallel
>> worlds”.
>
> But then by QM linearity, it is easy to prepare a superposition with
> orthogonal histories, like me seing a cat dead and me seeing a cat alive,
> when I look at the Schoredinger cat. Yes, decoherence makes hard for me to
> detect the superposition I am in, but it does not make it going away (unless
> you invoke some wave packet reduction of course)
>>
>> “Parallel worlds/histories” are just a popular name to describe a
>> superposition.
>>
>> In your dreams, maybe. There is a clear and precise definition of separate
>> worlds: they are orthogonal states that do not interact. The absence of
>> possible interaction means that they are not superpositions.
>
> That is weird.
> The branches of a superposition never interact. The point is that they can
> interfere statistically, if not there is no superposition, nor interference,
> only a mixture.
>
> There some to be some fluidity is the concepts of superposition and basis
> vectors inherent in this discussion. Any vector space can be spanned by a set
> of orthogonal basis vectors. There are an infinite number of such bases, plus
> the possibility of non-orthogonal bases given by any set of vectors that span
> the space. If the basis vectors are orthogonal, these basis vectors do not
> interact. But any general vector can be expressed as a superposition of these
> orthogonal basis vectors. (Orthonormal basis for a normed Hilbert space.)
>
> So the question whether the branches of a superposition can interact
> (interfere) or not is simply a matter of whether the branches are orthogonal
> or not. If we have a superposition of orthogonal basis vectors, then the
> branches do not interact. However, if we have a superposition of
> non-orthogonal vector, then the branches can interact.
>
> For example, the wave packet for a free electron is a superposition of
> momentum eigenstates (and position eigenstates). These momentum eigenstates
> are orthogonal and do not interact. The overlap function <p|p'> = 0 for all p
> not equal to p'. This is the definition of orthogonal states. But this does
> not mean that the wave packet of the electron is a mixture: It is a pure
> state since there is a basis of the corresponding Hilbert space for which the
> actual state is one of the basis vectors. (We can construct an orthonormal
> set of basis vectors around this vector.) On the other hand, the two paths
> that can be taken by a particle traversing a two-slit interference experiment
> are not orthogonal, so these paths can interact. So when the quantum state is
> written as a superposition of such paths, there is interference.
>
> Orthogonality is the key difference between things that can interfere and
> those that cannot. So if separate worlds are orthogonal, there can be no
> interference between them, and the absence of such interaction defines the
> worlds as separate.
What I use is the fact that when we have orthogonal states, like I0> and I1>, I
can prepare a state like (like I0> + I1>), and then I am myself in the
superposition state Ime>( I0> + I1>), Now, in that state, I have the choice
between measuring in the base {I0>, I1>} or in the base {I0> + I1>, I0> - I1>).
In the first case, the “parallel” history becomes indetectoble, but not in the
second case, so we have to take the superposition into account to get the
prediction right in all situations.
When I talk about some pure state, I mean it as an object considered before we
male a measurement on it. And I *assume* QM (without collapse) to be correct.
Bruno
>
> Bruce
>
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