On 6/12/2018 10:26 PM, Bruce Kellett wrote:
From: *Brent Meeker* <[email protected]>
On 6/12/2018 8:25 PM, Bruce Kellett wrote:
From: *Brent Meeker* <[email protected]
<mailto:[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?
Well, yes. Insofar as you can describe any vector in a linear space in
terms of any of the possible bases. But no. Not all of these
descriptions are the same -- what is given by the eigenvalues of one
operator will be a superposition of the eigenvalues of another
operator. In terms of position measurements, we get single dots on the
screen in the basis consisting of delta functions for positions along
the line.
I don't see that. Suppose I did a Fourier transform of the basis
consisting little bins across the screen. The indeed each spot on the
screen will be represented by a superposition of Fourier components, but
it will still be a spot in that representation. And the Schroedinger
eqn solution for the interference pattern on the screen will also be a
superposition of Fourier components.
Brent
Any other basis will give superpositions of the dots. Only one set of
basis vectors will describe what we see -- that is the basis that is
stable against decoherence.
Bruce
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!)
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