On 6/18/2018 10:21 PM, Bruce Kellett wrote:
From: *Brent Meeker* <[email protected]
On 6/17/2018 10:41 PM, Bruce Kellett wrote:
But the lens doesn't send one color to one photoreceptor and
another color to a different photorecptor. It focuses a spot of
light on several photorecptors and the one with the right pigment
fires its neuron. So it is energy detection.
But if you use a different position basis the lens will no longer
focus point objects to points on the retina.
I don't know enough about the physics of calorimeters as used in
HEP to comment here. But if temperature changes are measured by
bimetals or strain gauges, position comes into it in an essential
way.
Most work by measuring a voltage. But you miss the point. Those
position measurements are not essential in the QM sense. They are
just changing one classical value into another. Temperature is the
first classical level.
Fair enough. I suppose I am just very conscious of the fact that in
a different position basis all of this physics will be very
different. The classical universe will not look the same at all.
I guess I don't understand your idea of "position basis". My
understanding of linear algebra is that any basis that spans the
space can be used to represent any relation between structures. Why
should choosing a different basis make any difference to the physics
aside from the simplicity of its representation. It's just a
coordinate basis in Hilbert space. Or are you thinking of bases
different from position, e.g. momentum, energy, live/dead,...
Yes, there does seem to be a degree of miscommunication. I am not
think of different variables such as energy, momentum, or the like.
These are not different bases, they are different variables and they
inhabit different Hilbert spaces. So a change of base in one Hilbert
space does not take you to another space.
No, what I am considering is the possibility of different bases in a
single space, such as position space. If you assume an eignevector
interpretation of a set of basis vectors, then a different basis will
correspond to the eigenvalues of some different operator. It still
acts in the same, position, space, so it must be regarded as a
position operator, but it will have quite different physical
properties from the usual position operator that we use from classical
mechanics, where the eigenvectors are delta functions along the real line.
Because this will be a different operator, it will correspond to
different physics. For instance, if the position eigenvalues are
superpositions of delta functions, corresponding to superpositions of
different points, the point interactions of particles that we assume
in constructing the interaction Hamiltonian will be replaced by some
set of interactions between superpositions of points. This why I
suggest that the physics will be different. If the physics is the same
under this basis change, why is there any question about the preferred
basis? The point is that a change of basis does not mean that we
simply go to measure some other variable. I think Schlosshauer makes
this mistake, if I remember correctly; he seems to suggest that the
basis choice is between position or energy in most cases. That is just
wrong.
I think you're wrong about position operators. Sure, we usually think of
dividing space into little bins and a position operator has eigenvectors
that are 1 in some bin and zero in the other. But we could do the same
analysis in the Fourier transform of that space and the delta function
locations would be integrals over the wave numbers. It would be the
same physics. There would still be localized interactions. The
Hamiltonian would be written as an interaction of a superposition of
points, except they would all destructive interfere except at one
location. So the physics would be the same.
Consider the paradigmatic two slit experiment. The pattern you get on
the screen, which is predicted by the Schroedinger equation, is
described in your idea of position space by a lot of little bins that
have different degrees of probability, so if you put a detector there
you get a certain count rate. But that pattern on the screen is a
certain wavelet and if you transformed the Schroedinger equation to
wavelet space that whole pattern would be just one point in the space
and it would be the eigenvector of the two slit experiment.
The problem of the preferred basis arises in trying to explain why we
measure position of needles but not momentum or energy and why we don't
see superpositions of different needle positions. That's a question of
which measurement operator has eigenvectors stable against decoherence.
Not which basis we express the operator in. A basis doesn't have to
consist of the eigenvectors of the mesaurement operator, that's just
mathematically convenient and independent of the physics. No matter
what basis we use to write the operator in, it has the same eigenvectors.
Brent
Bruce
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