On 6/12/2018 8:25 PM, Bruce Kellett wrote:
From: *Brent Meeker* <[email protected]>
An isolated system has energy eigenvalues. But any realistic
macroscopic system is only going to conserve energy approximately. I
think energy eigenvalues are found in atoms and maybe molecules. But
larger systems (C60 Bucky balls?) tend to emit and absorb photons
that localize them in a position basis.
I am glad you said "a position basis" and not "the position basis" --
a mistake that is frequently made. Position is an operator in a high
dimensional Hilbert space, and there are an infinite number of
possible bases for this space, each corresponding to a different
operator in the space. Which one of these operators (and bases) is
"the" position basis? The answer from decoherence theory is that it is
the basis that is stable against environmental decoherence. But, as I
pointed out in a post on the 'Entanglement' thread, this is defined by
the operator that commutes with the interaction Hamiltonian. However,
the interaction Hamiltonian is usually defined in terms of point
particle interactions, so commutes with the position operator because
it contains that operator itself. So that particular definition of the
stable basis is circular -- any chosen operator in the position
Hilbert space would fit the bill provided it was used for both the
position measurement and the interaction Hamiltonian.
But is it a vicious circle? Aren't all the position bases going to be
physically equivalent?
Brent
We have to look elsewhere for the final explanation of "the preferred
basis". It might be that quantum gravity will give an explanation in
terms of the nature of quantum space-time. But it is possible that
Bohr was right all along, and the only final explanation is that the
"classical position" is the only stable basis, making the classical
prior to the quantum (which might not be an entirely satisfactory
outcome!)
Bruce
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