From: *Brent Meeker* <[email protected] <mailto:[email protected]>>
On 6/18/2018 10:21 PM, Bruce Kellett wrote:
From: *Brent Meeker* <[email protected] <mailto:[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.
I think that refers to regarding a change of basis as a simple
coordinate transformation, while retaining the actual operators of the
current (classical) theory, albeit in the form as transformed by the
change of coordinates. Clearly, that would not change the physics, it
would just change the way we describe it. The physics would not change
because you haven't changed what you mean by a measurement of position
because you still use the same position operator.
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.
That is regarding it as no more than a change of coordinates.
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.
No, I think that is wrong. It is not a matter of which variable we
measure, as in position, energy, momentum, or whatever. If that were the
case it would be the variable problem, not the basis problem. The basis
problem, if the words mean anything at all, must refer to the basis
which we use to expand our state vector when we make, say, a position
measurement. As you say, that is a question of which measurement
operator has eigenvectors that are stable against decoherence. It seems
to me that what is called into question is what we mean by a
measurement.....what operator in that Hilbert space corresponds to a
physical measurement of position. Each operator that we might choose has
a set of eigenvectors and eigenvalues. Given the measurement/eigenvalue
hypothesis of QM, if we have a different position operator, with
different eigenvectors and eigenvalues, the physics will necessarily be
different. We would expand that state in a different way:
|psi> = Sum_i a_i |i> or Sum_j b_j |j>
and the vectors and coefficients are different in the two cases. So a
measurement results in one of the |i> states, with probability |a_i|^2
in one case, and one the the |j> states, with probability |b_j|^2 in the
other case. Since both the states and the probabilities differ between
the cases, the physics must be different. This is more than just a
coordinate transformation, re-expressing the same physical operator in a
different form.
The problem of a preferred basis is then a question of which operator,
and which set of eigenvectors, is stable against environmental
decoherence. The same problem would arise for any measured quantity --
position, momentum, energy, time, and so on.
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.
True, but I am talking about using a different operator, and retaining
the eigenvector/eigenvalue correspondence for measurement results. In
other words, retaining the QM postulate that the result of a measurement
is an eigenvalue of the corresponding operator, and the measured state
ends up as the eigenvector corresponding to the observed eigenvalue. If
you retain the same operator, eigenvectors and eigenvalues, but merely
change the coordinates, then you get the same physics. But if you change
the operator to one with different eigenvectors/values, then you get
different physics.
As Zurek says:( arXiv:0707.2832)
"Popular accounts of decoherence and its role in the emergence of the
classical often start from the observation that when a quantum system
interacts with some environment 'phase relations in the system are
lost'. This is a caricature, at best incomplete if not misleading: It
begs the question: 'Phases between what?' This in turn leads directly to
the main issue addressed by einselection: 'What is the preferred
basis?'. This question is often muddled in 'folklore' accounts of
decoherence"
Bruce
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