Re: What Superposition doesn't imply ...

[Brent Meeker]

On 1/16/2025 3:04 AM, Alan Grayson wrote:


On Wednesday, January 15, 2025 at 6:51:28 PM UTC-7 Alan Grayson wrote:

    On Wednesday, January 15, 2025 at 6:44:27 PM UTC-7 Brent Meeker wrote:




        On 1/15/2025 4:55 PM, Alan Grayson wrote:


        On Wednesday, January 15, 2025 at 5:15:35 PM UTC-7 Brent
        Meeker wrote:




            On 1/15/2025 1:39 PM, Russell Standish wrote:
            What you are talking about is known as the preferred
            basis problem. This has been discussed on this list
            before. My own take on this is that you can't ignore the
            observer. In any physical situation, an observer chooses
            some measurement apparatus (thereafter you can sweep the
            observer under the carpet, and focus on the measurement
            apparatus). The measurement apparatus entangled with the
            system under question has the dynamics that tensor
            product of measuring apparatus state with that of the
            system evolves to be diagonal in some basis, aka
            "einselection". And that is the origin of the preferred
            basis. In the multiverse, there will also be other
            observers choosing different apparati eg ones that
            select a complementary basis (eg momentum where the
            first chooses to measure position). These will have a
            different set of preferred basis. There is only a
            problem if you try to ignore the existence of observers
            and measuring devices. Cheers On Wed, Jan 15, 2025 at
            11:58:33AM -0800, Alan Grayson wrote:
            It's easy to show that a Superposition does NOT imply
            that a system represented by a linear sum of a pure set
            of basis vectors, is in all of those states
            simultaneusly.This follows from the fact that the WF is
            an element of a vector space, a Hilbert space, and in
            vector spaces there is no unique set of basis vectors.
            IOW, any set of basis vectors can represent the WF of a
            system, and if we claim the system is in all states of
            some superposition, it must also be in all states of
            any other superposition. 
            If it's in a pure state then that is single vector in
            Hilbert space.  So there is a basis
            that includes that vector and then the state has a single
            component in that basis.
            Of course there is no way to measure in that basis
            without already knowing what
            what it is.

            Brent

         Generally speaking, isn't a superposition a linear sum of
        pure states? AG

        Right.  And a linear sum of vectors is a vector.

        Brent

    *
    *
    *If it can be proven what I've initially stated about a
    superposition, why is it necessary to consider entangement of
    experimenter and*
    *apparatus, when the result follows directly from the properties
    of a vector space? AG*


*Another question I have about superposition is this; if the wf for 
some system is written in a momentum basis, do other momentum bases *
*exist for expressing the wf? If so, is the set of momentum bases 
infinite? AG*
*It's Hilbert space.  For finite dimensions it's just a vector space.  
Forget about bases, vectors are things that at independent of bases.  
Bases are arbitrary.

Brent*

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