QM Measurement Problem

[Martin Hogbin]

More of a mad idea.

Some while ago I posted the suggestion that uncertainty in QM might be
conserved in some way. The basis for this suggestion was this idea.

Suppose we have a system in which a particle is described by some
wavefunction. Before a measurement is made there is a level of
uncertainty as to its position. What happens a measurement is made?
Let us say that the result of that measurement is indicated on an
analogue meter. If the original uncertainty were to remain that would
result in an uncertainty in the position of the meter needle. In other
words a small uncertainty (of the microscopic particle) would be changed
into a large one (of the meter needle). So what actually happens is
that uncertainty is conserved and a measurement is made.

What I am trying to say is that a measurement could be considered as
anything which would turn a small uncertainty (however that is
quantified) into a larger one. I realise that the situation is more
complicated than this in that uncertainty need to be applied to pairs of
conjugate variables, but I have picked just one variable to try to
illustrate the point.

This approach has the great advantage of removing issues about who or
what has ability to make the measurement (computer, human, cat). Any
thing that would turn the quantum uncertainty into a larger one is a
measurement.

Does anyone see any value in this approach? How close is it to quantum
decoherence? If uncertainty (in some form) is conserved, what kind if
symmetry would this imply?

--
Martin Hogbin

[Doug Sweetser]

Hello Martin:

Unfortunately I see no value in what you wrote. The reason is that
the uncertainty principle is about the relationship between two
qualitatively different types of measurements. You referred only to
position. The conjugate measurement for position is momentum.
Position can be measured to an arbitrary degree of precision: spend
more money, get a better measure. There is no limit to how precise
one can be. Use an electron microscope to watch the needle move.

The same can be said for measuring momentum. We can know how much
punch a punch has to arbitrary precision. One of the big costs at any
atom smasher is measuring the energy and momentum of particles
produced.

What we cannot know is both the position and the momentum in the same
direction. Position in x and momentum in y are not conjugate
variables, so they can be measured together to arbitrary precision.
This _pair_ of measurements is governed by the uncertainty principle.

Although position and momentum are must often used in the popular
literature, there are all kinds of other neat pairs of conjugate
variables. I bet there is a theorem out there that all measurements
have a conjugate. Anyone recall the conjugate for number of photons?
It is the light intensity?

Doug

[Mentz114]

Anyone recall the conjugate for number of photons?
It is the light intensity?

The conjugate of the number density is the phase - but only with some caveats, which I will look up.

[Martin Hogbin]

"Doug Sweetser" <dougsweetser@gmail.com> wrote in message
news:5ce7bdc7-ca37-4323-9b78-ec08afe6b6b4@s37g2000prg.googlegroups.com...
> Hello Martin:
>
> Unfortunately I see no value in what you wrote. The reason is that
> the uncertainty principle is about the relationship between two
> qualitatively different types of measurements. You referred only to
> position. The conjugate measurement for position is momentum.
> Position can be measured to an arbitrary degree of precision: spend
> more money, get a better measure. There is no limit to how precise
> one can be. Use an electron microscope to watch the needle move.
>
> The same can be said for measuring momentum. We can know how much
> punch a punch has to arbitrary precision. One of the big costs at any
> atom smasher is measuring the energy and momentum of particles
> produced.
>
> What we cannot know is both the position and the momentum in the same
> direction. Position in x and momentum in y are not conjugate
> variables, so they can be measured together to arbitrary precision.
> This _pair_ of measurements is governed by the uncertainty principle.
>
> Although position and momentum are must often used in the popular
> literature, there are all kinds of other neat pairs of conjugate
> variables. I bet there is a theorem out there that all measurements
> have a conjugate. Anyone recall the conjugate for number of photons?
> It is the light intensity?

Yes, I know about conjugate variables, that is why I mentioned the
term, but I cannot see how to deal with this fact so I just ignored it
for the moment.

I am trying to find an objective definition of 'measurement', one that
need not involve physicists, other humans, or cats. I want to be able
say that this or that experimental setup constitutes a 'measurement'
regardless of whether anybody looks at the results. It seems to me that
the basic function of a measurement is to turn a small uncertainty into
a large one. The questions I wonder are:

How can we quantify the uncertainty?
To what system does it apply?

As an aside, if Schrodinger's cat is replaced by a physicists, who
I guess we must deem able to make a measurement, then the
situation gets even more weird. Presumably the physicist knows
that he is alive (if that is the case) and the measurement is thus made.
If on the other hand he is dead then he cannot make the measurement
to that effect and he therefore must remain half-dead/ half-alive,
except that we have just said that he is dead. Weird!

--
Martin Hogbin

[Keith Blow]

Doug Sweetser wrote:
> have a conjugate. Anyone recall the conjugate for number of photons?
> It is the light intensity?
>
> Doug
>
The conjugate to photon number is phase. This brings up a whole bunch of
problems associated with the fact that the phase is only defined modulo
2pi so the maximum uncertainty is 2pi. Another one is what the
eigenstates are that satisfy the minimum uncertainty condition.

--
Keith Blow

[Doug Sweetser]

Hello Martin:

You are not at liberty to ignore the issue of complementarity as it is
central to understanding the uncertainty princple.

The physics of the uncertainty principle applies to physical systems.
A vanishingly small number of these observations are observed by
physicists. Whether we look at the position and momentum or not is
not relevant - the uncertainty principle governs the relationship
between all conjugate variables.

The uncertainty principle arises from the properties of complex
numbers, which unlike the real numbers, is not a totally ordered set.
I was lucky enough to attend a quantum mechanics class where the
professor showed that the equation of the uncertainty principle can be
derived from properties of complex numbers. The lecture made quite an
impression on me. I recreated his talk, but applied it to quaternions
which are 3 complex numbers that share the same real, here:

http://www.theworld.com/~sweetser/quaternions/quantum/uncertainty/uncertainty.html

The notes should make clear: the product of the variation of the
measurements of two conjugate variables being greater than their
commutator has nothing to do with amplifying a small uncertainty into
a large one. I do encourage people to take guesses at new
interpretations, and hope they accept that the odds those ideas are
correct are vanishingly small.


> How can we quantify the uncertainty?
> To what system does it apply?

Look up the equation and apply to all conjugate variables.

As an aside, I avoid all discussions of the cat and work with the
details of the equations. The web page referenced above has an
impolite number of equations, but that is the way real physics is
versus pop physics. The equations are concise, but the word stories
are muddled.

Doug

[Martin Hogbin]

"Doug Sweetser" <dougsweetser@gmail.com> wrote in message
news:b9a6e7f5-61d2-4cc7-aef5-b68154ba7fbf@e25g2000prg.googlegroups.com...
> Hello Martin:
>
> You are not at liberty to ignore the issue of complementarity as it is
> central to understanding the uncertainty princple.
>
> The physics of the uncertainty principle applies to physical systems.
> A vanishingly small number of these observations are observed by
> physicists. Whether we look at the position and momentum or not is
> not relevant - the uncertainty principle governs the relationship
> between all conjugate variables.

Yes, I know.

> The uncertainty principle arises from the properties of complex
> numbers, which unlike the real numbers, is not a totally ordered set.
> I was lucky enough to attend a quantum mechanics class where the
> professor showed that the equation of the uncertainty principle can be
> derived from properties of complex numbers. The lecture made quite an
> impression on me. I recreated his talk, but applied it to quaternions
> which are 3 complex numbers that share the same real, here:
>
> http://www.theworld.com/~sweetser/quaternions/quantum/uncertainty/uncertainty.html

An interesting page, but it is about the formalism of QM and
the application of quaternions it does not address the 'measurement
problem'.


> The notes should make clear: the product of the variation of the
> measurements of two conjugate variables being greater than their
> commutator has nothing to do with amplifying a small uncertainty into
> a large one.

I think you are misunderstanding my suggestion. The large uncertainty
that I am referring to is that that the position (and momentum) of the
indicating needle would have in my thought experiment, if the position
(and momentum) of the measured particle remained uncertain.

> I do encourage people to take guesses at new
> interpretations, and hope they accept that the odds those ideas are
> correct are vanishingly small.
>
Yes, I realise that my chances of coming up with anything new and
useful are small but I like to keep trying.


--
Martin Hogbin

[Mentz114]

The uncertainty principle arises from the properties of complex
numbers, which unlike the real numbers, is not a totally ordered set.
This is not physics, Doug. How can the properties of an abstract mathematical structure be predictive of something as fundemental as the UP ?

The uncertainty principle arises from deep physical causes, not properties of ordered/unordered sets.

[Gerry Quinn]

In article <BISdnb_XWbSddCzanZ2dneKdnZydnZ2d@bt.com>, goatREMOVETHIS123
@hogbin.org says...

> I am trying to find an objective definition of 'measurement', one that
> need not involve physicists, other humans, or cats. I want to be able
> say that this or that experimental setup constitutes a 'measurement'
> regardless of whether anybody looks at the results. It seems to me that
> the basic function of a measurement is to turn a small uncertainty into
> a large one. The questions I wonder are:
>
> How can we quantify the uncertainty?
> To what system does it apply?
>
> As an aside, if Schrodinger's cat is replaced by a physicists, who
> I guess we must deem able to make a measurement, then the
> situation gets even more weird. Presumably the physicist knows
> that he is alive (if that is the case) and the measurement is thus made.
> If on the other hand he is dead then he cannot make the measurement
> to that effect and he therefore must remain half-dead/ half-alive,
> except that we have just said that he is dead. Weird!

But the weirdness comes precisely from your insistence that there must
be an "objective" definition of measurement, or, put another way, that
state vector resolution is a physical process that occurs at a
particular point in spacetime!

[The necessity of placing it at a particular point in spacetime, in
fact, seems to be another problem with the concept of physical state
vector resolution that I haven't seen mentioned before. Since quantum
systems are often extended in spacetime, the process of state vector
resolution, however we understand it, must be extended also. And this
seems to fit very poorly with any simplistic approach to the issue.]

Personally I think that Schrodinger's Cat leads naturally to the concept
of decoherence. It can perhaps be seen better with another gedanken
which I will call the Simplified Schrodinger's Cat.

In the Simplified Schrodinger's Cat Experiment, we simply put a cat in a
box, close the lid for a period, and open it again. There are a large
variety of things we might now observe; the cat may be asleep or awake,
it may have scratched or otherwise left its mark in one or more places
inside the box. And it's obvious that no meaningful quantum
superpositions can be observed between any two distinct states of this
kind. The cat has apparently been generating entropy via interactions
with itself and anything else on the inside of the box - or at least the
portion of the wave function of the universe covering the region of
spacetime where the box was closed has evolved in such a way as to
produce an equivalent result.

Wherever we choose to define measurement - inside the box, or when it is
opened - it makes no difference to the physics. And thus, it would
appear, state vector resolution is not physical, unless it operates
according to some criterion we do not currently recognise.

- Gerry Quinn

[Doug Sweetser]

Hello Gerry:

I found your "Simplified Schrodinger's Cat Experiment" amusing to
think about :-) It opened up a question: why is the system not
really like quantum mechanics? My reply is that cats have all kinds
of unique identifiable parts. With a group of electrons or a vast
pool of photons, one cannot pick out one in particular, and say, the
electron left of Ginger is Gilligan. No amount of money can label two
electrons. We can tell if they happen to be in different states, but
the electrons could swap positions while we were not looking, and we
would not be able to tell.

In my SSCE, there would be 1000 boxes, each with 1000 Siamese cat
clones. In about 500 boxes, the cat has died, in the other 500, the
cat clone is alive. Take a picture of all cats in all boxes with your
12.5 mega pixel Nikon D300, import into the Gimp, overlay all the
images and average. One thing you notice immediately: the cat looks
like a gas! Superimposing this many images makes what was once so
solid look very flimsy indeed. I did this for animations of a simple
harmonic oscillator which was quite cool (URL at end). You could tell
from the ghostly image that many of the states of the cloned cat in a
box have a dead cat in it.

Now you do the experiment of picking one of these 1000 boxes, which is
an act of observation, not of action. You open a box - you cannot
number it and say it is box 27 because the boxes are all
indistinguishable - and see a dead cat. Repeat 50 times, and about
half the time the cat is dead, half the time alive. The observation
is NOT killing the cat. The superpositon does look half alive/half
dead because that is exactly what goes into the Gimp from the D300.

Doug

Simple Harmonic Oscillators: Visualizing Classic and Quantum
http://www.youtube.com/watch?v=efYhDxm1m-g

[Gerry Quinn]

In article <b4626a34-2d76-4127-b325-
6aa060aa578e@n75g2000hsh.googlegroups.com>, dougsweetser@gmail.com
says...
> Hello Gerry:
>
> I found your "Simplified Schrodinger's Cat Experiment" amusing to
> think about :-) It opened up a question: why is the system not
> really like quantum mechanics? My reply is that cats have all kinds
> of unique identifiable parts. With a group of electrons or a vast
> pool of photons, one cannot pick out one in particular, and say, the
> electron left of Ginger is Gilligan. No amount of money can label two
> electrons. We can tell if they happen to be in different states, but
> the electrons could swap positions while we were not looking, and we
> would not be able to tell.

Well - even leaving the Pauli Exclusion Principle out of it - that's not
really the difference. You could say the same of atoms. Yet a solid
structure built out of identical atoms behaves classically, and even
individual atoms in such a structure can behave in some ways
classically. Suppose you took a flat facet of an iron crystal and used
a scanning tuneling microscope to place iron atoms on it so they spelt
your name. Those iron atoms will be persistent; they can be imaged
repeatedly; they don't do anything crazy at all. They behave
classically, at least insofar as their position is concerned.

The difference, of course, is that the atoms aren't in a 'cloud'; they
are in a solid, and their interaction with their environment is such as
to lead to rapid decoherence. They are entangled with each other in
such a fashion that the probability of observing a superposition state
is infinitesimal.

This also, obviously, applies to a cat. And cats also generate entropy,
which can be considered a measure of the number of measurements carried
out and recorded by the system. So while you might, if you handled the
system very carefully, be able to observe some quantum superposition
properties of the iron atoms, you have no chance with the cat.

> In my SSCE, there would be 1000 boxes, each with 1000 Siamese cat
> clones. In about 500 boxes, the cat has died, in the other 500, the
> cat clone is alive. Take a picture of all cats in all boxes with your
> 12.5 mega pixel Nikon D300, import into the Gimp, overlay all the
> images and average. One thing you notice immediately: the cat looks
> like a gas! Superimposing this many images makes what was once so
> solid look very flimsy indeed. I did this for animations of a simple
> harmonic oscillator which was quite cool (URL at end). You could tell
> from the ghostly image that many of the states of the cloned cat in a
> box have a dead cat in it.

I assume you are not actually taking a picture of each cat, which would
correspond to an observation, negating the point of the experiment. So
your picture corresponds to an ensemble of expected classical results.
I don't see where quantum statistics come into it.

> Now you do the experiment of picking one of these 1000 boxes, which is
> an act of observation, not of action. You open a box - you cannot
> number it and say it is box 27 because the boxes are all
> indistinguishable - and see a dead cat. Repeat 50 times, and about
> half the time the cat is dead, half the time alive. The observation
> is NOT killing the cat. The superpositon does look half alive/half
> dead because that is exactly what goes into the Gimp from the D300.

I feel you are missing the point here. Your combined image isn't a
picture of a quantum superposition state; it is a picture of 1000
classical states, classically superimposed on each other (with additive
probability).

(And you also can't make the boxes indistinguishable, for reasons
similar to those already discussed.)

- Gerry Quinn

[Salviati]

"Doug Sweetser" <dougsweetser@gmail.com> wrote in
news:b4626a34-2d76-4127-b325-6aa060aa578e@n75g2000hsh.googlegroups.com...
>...- you cannot
> number it and say it is box 27 because the boxes are all
> indistinguishable - and see a dead cat.

In mathematical terminology we used to say: The reals are uncountable. The
perhaps first one who clearly expressed that the relations >, =, and < do
not fit infinite quantities was Salviati alias Galileo Galilei when
commenting on bijection.

[Salviati]

"Martin Hogbin" <goatREMOVETHIS123@hogbin.org> wrote ..
>
> "Doug Sweetser" <dougsweetser@gmail.com> wrote in message
> news:b9a6e7f5-61d2-4cc7-aef5-b68154ba7fbf@e25g2000prg.googlegroups.com...

>> The uncertainty principle arises from the properties of complex
>> numbers, which unlike the real numbers, is not a totally ordered set.

What about the possiblity to attribute the idea of having two totally
ordered sets at a time on a simple real-valued pair of conjugate exclusively
positive quantities x and y=1/x ? Let be F(x) the cosine transform of f(y).

[Doug Sweetser]

Hello Gerry:

> Well - even leaving the Pauli Exclusion Principle out of it

The Pauli Exclusion Principle is about states that indistinguishable
particles with half integrable spin can occupy. I was addressing the
indistinguishable part, not the behavior of half integral spin
particles. It is safe to leave Pauli for another discussion.


> You could say the same of atoms.

I would to be logically consistent. Much about atoms can be described
by quantum mechanics, much can be described by classical physics.
Nature keeps multiple accounting books open on the same objects.

> Yet a solid structure built out of identical atoms behaves classically, and even
> individual atoms in such a structure can behave in some ways
> classically. Suppose you took a flat facet of an iron crystal and used
> a scanning tuneling microscope to place iron atoms on it so they spelt
> your name. Those iron atoms will be persistent; they can be imaged
> repeatedly; they don't do anything crazy at all. They behave
> classically, at least insofar as their position is concerned.

I don't see the point. It looks like you are alluding to the
uncertainty principle, which is not about measuring the certainty of
position. Rather the uncertainty is about measuring the position of
two conjugate measurements, say position and momentum, and recognizing
a constraint on the variation of both measurements. That would apply
to this iron crystal. If this is a well ordered crystal, you probably
could collect data on the photo electric effect. The same collection
of atoms knows how to do both classical and quantum mechanics.

> The difference, of course, is that the atoms aren't in a 'cloud'; they
> are in a solid, and their interaction with their environment is such as
> to lead to rapid decoherence. They are entangled with each other in
> such a fashion that the probability of observing a superposition state
> is infinitesimal.


If I read a little into this (in other words, you did not say this
directly, so I apologize in advance if needed), it sounds like the
'cloud' does not have information. In quantum mechanics, the
probability distribution represented by the wave function is the best
and most complete information one can have about a system. It is the
sum of all possible paths. The superposition cannot be seen because
it is a composition of data that can be measured.

> This also, obviously, applies to a cat. And cats also generate entropy,
> which can be considered a measure of the number of measurements carried
> out and recorded by the system. So while you might, if you handled the
> system very carefully, be able to observe some quantum superposition
> properties of the iron atoms, you have no chance with the cat.

Entropy is not relevant to this discussion. I created the
superposition of a live and dead cat here:

http://picasaweb.google.com/dougsweetser/Superposition/photo#5168676570581305906


> I assume you are not actually taking a picture of each cat, which would
> correspond to an observation, negating the point of the experiment.

No, that is incorrect. We have a system that produces about 50% dead
Siamese cats clones every time we have investigated. One time, we
make all these observations, collect all the data, and from that data,
construct the superposition. We have a system that can repeatably
generate indistinguishable collections of dead cats.

It appears like you are using the phrase "classical result" for what
might also be referred to as the collapse of the wave function. I
don't like either term, both sounding to active.


> So your picture corresponds to an ensemble of expected classical results.
> I don't see where quantum statistics come into it.


I am only trying to get at superposition, not even and odd spin
statistics.


> I feel you are missing the point here. Your combined image isn't a
> picture of a quantum superposition state; it is a picture of 1000
> classical states, classically superimposed on each other (with additive
> probability).
>
> (And you also can't make the boxes indistinguishable, for reasons
> similar to those already discussed.)


I don't recall in classical physics where folks deal with the
information of superpositions of states. That sounds far more like
quantum mechanics than classical mechanics, where one has the video of
the cat, either live or dead, but certainly not both. In my
superposition photo, the cat is unambiguously live and dead, an idea
that is suppose to be too odd to understand. I get it. I also
understand why you don't accept that I get it, so we can politely
disagree.

Doug

[Doug Sweetser]

Hello Salviati:

I like this quote:

> The reals are uncountable.

That had deep implications for our description of Nature. What those
implications are can be confusing.

> What about the possibility to attribute the idea of having two totally
> ordered sets at a time on a simple real-valued pair of conjugate exclusively
> positive quantities x and y=1/x ? Let be F(x) the cosine transform of f(y).

This issue is that you are making up the rule for doing the ordering.
If someone else wanted to define F(x) as the sine transform of f(y),
the ordering would be different. A simpler approach would be to order
by the real value of x, and if there was a tie, order by the imaginary
y. An equally valid approach would involve switching the role of x
with y.

One way to view this from a math perspective is that quantum mechanics
uses the norm, z* z, which has a least lower bound of zero. For me, I
look at that least lower bound as the vacuum - there can be nothing
less than nothing. Take two numbers, z and z', and the product z* z'
does not have a least lower bound, but (z* z')* z* z' does. There is
nothing difficult about this math, but I appreciate it is a stretch to
see a connection to quantum mechanics, since quantum mechanics is
about the physical world, and this is a math observation.

Doug

[p.kinsler@ic.ac.uk]

Keith Blow <kb@somewhere.home> wrote:
> The conjugate to photon number is phase. This brings up a whole bunch of
> problems associated with the fact that the phase is only defined modulo
> 2pi so the maximum uncertainty is 2pi. Another one is what the
> eigenstates are that satisfy the minimum uncertainty condition.

Phase properties of the quantized single-mode electromagnetic field
D. T. Pegg, S. M. Barnett
Phys. Rev. A 39, 1665 - 1675 (1989)
http://prola.aps.org/abstract/PRA/v39/i4/p1665_1

On the Hermitian optical phase operator
S. M. Barnett; D. T. Pegg
J. Mod. Opt. 36, 7 (1989)
http://www.informaworld.com/smpp/content~content=a713822510

--
---------------------------------+---------------------------------
Dr. Paul Kinsler
Blackett Laboratory (QOLS) (ph) +44-20-759-47520 (fax) 47714
Imperial College London, Dr.Paul.Kinsler@physics.org
SW7 2BW, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/

[Gerry Quinn]

In article <27ad8097-7a65-4dea-be09-866700d5b6b0@
41g2000hsc.googlegroups.com>, dougsweetser@gmail.com says...
> Hello Gerry:

> > Yet a solid structure built out of identical atoms behaves classically, and even
> > individual atoms in such a structure can behave in some ways
> > classically. Suppose you took a flat facet of an iron crystal and used
> > a scanning tuneling microscope to place iron atoms on it so they spelt
> > your name. Those iron atoms will be persistent; they can be imaged
> > repeatedly; they don't do anything crazy at all. They behave
> > classically, at least insofar as their position is concerned.
>
> I don't see the point. It looks like you are alluding to the
> uncertainty principle, which is not about measuring the certainty of
> position. Rather the uncertainty is about measuring the position of
> two conjugate measurements, say position and momentum, and recognizing
> a constraint on the variation of both measurements. That would apply
> to this iron crystal. If this is a well ordered crystal, you probably
> could collect data on the photo electric effect. The same collection
> of atoms knows how to do both classical and quantum mechanics.

I was responding to your proposal that cats were special because they
were made up of non-identical parts which could swap position if we were
not looking. As pointed out above, collections of identical parts can
behave classically too, so long as the parts are sufficiently entangled
with each other or the environment in general.

And yes - the uncertainty principle does apply to the crystal. But the
interesting thing is that it does not apply, at least naively, to the
individual atoms. A particular atom, say the one creating the top serif
of the 'D', can consistently be observed to be in the same place, and it
is not moving anywhere. There is a real sense in which its position is
known exactly, and its momentum is zero. It is living in the classical
world, precisely because of its entanglement with the other atoms.

> > The difference, of course, is that the atoms aren't in a 'cloud'; they
> > are in a solid, and their interaction with their environment is such as
> > to lead to rapid decoherence. They are entangled with each other in
> > such a fashion that the probability of observing a superposition state
> > is infinitesimal.
> > If I read a little into this (in other words, you did not say this
> directly, so I apologize in advance if needed), it sounds like the
> 'cloud' does not have information. In quantum mechanics, the
> probability distribution represented by the wave function is the best
> and most complete information one can have about a system. It is the
> sum of all possible paths. The superposition cannot be seen because
> it is a composition of data that can be measured.

I didn't say anything like that. I do not know how to phrase what I did
say better. The atoms in the solid are strongly entangled with others
in their environment - the electrons in your vaguely-described 'cloud',
presumably, are not.

> > This also, obviously, applies to a cat. And cats also generate entropy,
> > which can be considered a measure of the number of measurements carried
> > out and recorded by the system. So while you might, if you handled the
> > system very carefully, be able to observe some quantum superposition
> > properties of the iron atoms, you have no chance with the cat.
>
> Entropy is not relevant to this discussion.

It is, IMO. It is another special feature of cats which brings them
further away from the quantum world.

> I created the
> superposition of a live and dead cat here:
>
> http://picasaweb.google.com/dougsweetser/Superposition/photo#5168676570581305906

That's not what a quantum superposition state of Schrodinger's cat would
look like. That is two possible classical states superimposed. (Two
very distinct states, in fact, selected from a myriad possibilities.)

You could semi-plausibly have a superposition state that looked like a
cat that you could not tell was alive or dead. It would instantly
collapse into one or the other, or more likely into a very sick cat.

You can make a case that a picture of all classical possibilities
superimposed describes the wave function of the cat at a particular
time. But this wave function has *already* in the process of its own
evolution decohered into an ensemble of non-interfering, effectively
classical states. Opening the box just continues this process.

> > I assume you are not actually taking a picture of each cat, which would
> > correspond to an observation, negating the point of the experiment.
>
> No, that is incorrect. We have a system that produces about 50% dead
> Siamese cats clones every time we have investigated. One time, we
> make all these observations, collect all the data, and from that data,
> construct the superposition. We have a system that can repeatably
> generate indistinguishable collections of dead cats.

So you are taking the classical results observed after opening each box,
and superimposing them. Just what I said. No picture of an observed
superposition state. No indication of interference between dead and
alive states.

> It appears like you are using the phrase "classical result" for what
> might also be referred to as the collapse of the wave function. I
> don't like either term, both sounding to active.

Classical result = what you get when you open the box and see a dead cat
or a live cat. Nothing more.

> > So your picture corresponds to an ensemble of expected classical results.
> > I don't see where quantum statistics come into it.
>
> I am only trying to get at superposition, not even and odd spin
> statistics.

So am I. I am not talking about spin statistics. The point is that
quantum superposition states imply interference between the different
possible results. Superimposing a lot of non-interfering classical
states does not give such a state.

> > I feel you are missing the point here. Your combined image isn't a
> > picture of a quantum superposition state; it is a picture of 1000
> > classical states, classically superimposed on each other (with additive
> > probability).
> >
> > (And you also can't make the boxes indistinguishable, for reasons
> > similar to those already discussed.)

> I don't recall in classical physics where folks deal with the
> information of superpositions of states.

But everyone who deals with probability produces diagrams similar in a
sense to your superimposed diagrams. If you want to analyse a dice
game, for example, you start by (notionally) drawing a picture of all
the different ways the dice might land. There's nothing non-classical
here. Classical thermodynamics involves similar scenarios.

> That sounds far more like
> quantum mechanics than classical mechanics, where one has the video of
> the cat, either live or dead, but certainly not both. In my
> superposition photo, the cat is unambiguously live and dead, an idea
> that is suppose to be too odd to understand. I get it. I also
> understand why you don't accept that I get it, so we can politely
> disagree.

"Unambiguously live and dead"? What do you mean by that? You could say
that both live and dead cats exist in Everett's 'multiverse', and that
your diagram is a picture of the multiverse, I suppose. What it is not
is a superposition state in the sense of the word that is normally used.
- Gerry Quinn

[Salviati]

"Doug Sweetser" <dougsweetser@gmail.com> schrieb im Newsbeitrag
news:8b95db5f-b7dc-46bd-bd77-4624dea1c00a@c33g2000hsd.googlegroups.com...
> Hello Salviati:
>
> I like this quote:
>
>> The reals are uncountable.
>
> That had deep implications for our description of Nature. What those
> implications are can be confusing.

Mathematicians would 'correct' me: The set of reals is uncountable. I am
arguing: Each single irrational number can be thought to be indefinitely
long. And I add: Uncertainty of a pair of conjugate variables originates
there. Any analog measurement is uncertain in so far, one can increase
acuity at will but one will never reach an absolutely exact value. I
consider you correct: Uncertainty relation just makes this obvious. The
mistake is as old as are Zeno's turtle and Buridan's donkey.

>> What about the possibility to attribute the idea of having two totally
>> ordered sets at a time on a simple real-valued pair of conjugate
>> exclusively
>> positive quantities x and y=1/x ? Let be F(x) the cosine transform of
>> f(y).
>
> This issue is that you are making up the rule for doing the ordering.

G. Cantor has shown: There are not two joint but independent dimensions
of totally ordered infinite sets. If x is known then y is also known. I
disagree with Dedekind and Cantor: We cannot even make up mathematics at
will if we expect reasonable results.

Ordering of a physical quantity can be arbitrary like the Celsius scale
or natural. Elapsed time and radius have an absolute zero.

> If someone else wanted to define F(x) as the sine transform of f(y),
> the ordering would be different.

Of course, this is true if we consider x a relative quantity.

>A simpler approach would be to order
> by the real value of x, and if there was a tie, order by the imaginary
> y. An equally valid approach would involve switching the role of x
> with y.

Even if we perform complex Fourier transform, only the functions of x
and y are not real.

Switching the role of x with y was exactly what Heisenberg,
Schroedinger, and Dirac did.

> One way to view this from a math perspective is that quantum mechanics
> uses the norm, z* z,

Multiplication by the complex conjugate intended getting rid of
'unphysical' imaginary components. Nobody realized that one-sidedness
got lost. Just Weyl worried, and v. Neumann gave up Hilbert space.

[Salviati]

Doug Sweetser did not yet get an answer to his question
> Anyone recall the conjugate for number of photons?
> It is the light intensity?

Let me look at the issue by comparison with time and frequency. Both are
usually considered continuous quantities. Nonetheless, there are pretty
discrete nearly hyperbolic lines linking the pair, cf.
http://home.arcor.de/eckard.blumschein.M283.html I will add some new results
perhaps as M284 next week.
We used to be sure that photons are countable: NoN, one, two, three,...
nothing in between. Likewise we could count frequencies.

[Doug Sweetser]

Hello Salviati:

> Mathematicians would 'correct' me: The set of reals is uncountable. I am
> arguing: Each single irrational number can be thought to be indefinitely
> long. And I add: Uncertainty of a pair of conjugate variables originates
> there.

The math sounds right, the application to physics does not. The
reason is that any single measurement has this property, even
measurements made for classical systems. The proof of the uncertainty
principle I saw has to do with the properties of complex numbers, not
the properties of the reals to which you referred. I recall reading
in a book by Stephen Adler that the quantum numbers done over the
field of real numbers would be quite dull, no quantum interference.

Doug

[Doug Sweetser]

Hello Gerry:

You wrote:

> A particular atom, say the one creating the top serif
> of the 'D', can consistently be observed to be in the same place, and it
> is not moving anywhere. There is a real sense in which its position is
> known exactly, and its momentum is zero. It is living in the classical
> world, precisely because of its entanglement with the other atoms.

One certainly can do a series of experiments where the atoms are
proven by observation to be sitting right on top of the 'D'. One
could do another series of experiments to show that the momentum is
zero. One cannot do a single experiment to show that in a measurable
sense, "its position is known exactly, and its momentum is zero"
without violating the uncertainty principle, the variation of the
measurement of the position x times the variation in the measurement
of the momentum px must be greater than the super tiny hbar.

This was an instructive comment:

> The atoms in the solid are strongly entangled with others
> in their environment - the electrons in your vaguely-described 'cloud',
> presumably, are not.

The cloud is both very precise and reproducible. There are many
systems that with their superposition of states have a location where
there is zero probability to find the atom. An experiment can be set
up to measure the probability distribution of the system over space,
and we find that there are places with zero probability. We label
this quantum interference. Unfortunately, we bring with us the notion
of classical interference, where one thing gets together with another
thing destructively. That is not the way quantum systems work -
everything is independent.

> You could semi-plausibly have a superposition state that looked like a
> cat that you could not tell was alive or dead. It would instantly
> collapse into one or the other, or more likely into a very sick cat.
...
> So you are taking the classical results observed after opening each box,
> and superimposing them. Just what I said. No picture of an observed
> superposition state. No indication of interference between dead and
> alive states.

This indicates we are not communicating so well on these issues, which
is not uncommon. There is no need for a sick cat state. Nor is there
a need to show interference between the dead and alive states. As
soon as I discuss making a measurement of anything, you slap the label
"classical result", a behavior I find mystifying. The CCD camera at
the end of a two slit interference experiment would appear to fit this
notion of classical result since the signals are either on or off.

So here is your definition:

> Classical result = what you get when you open the box and see a dead cat
> or a live cat. Nothing more.

Unfortunately, I don't get what you mean. For me, what classical
physics is about is our ability to watch a system, say a live cat,
evolve in time. We can watch a cat go from alive to sick to dead over
a period of observation.

The SSCE I described ("Simplified Schrodinger's Cat Experiment"),
there is no observation you can ever do with the cat transitioning in
time from live to dead. One gets one or the other. The best we can
do to summarize the results is a picture like I provided.

Neat, I see a connection to quantum interference. Earlier I had said
there is no "sick cat" state. That is a state the classical physicist
would expect to see, part of the transition in time from live to
dead. In the precise, repeatable system I set up with a thousand cat
clones, not a single one was sick. It is the omission of expected
states that many find troubling about quantum mechanics.

> "Unambiguously live and dead"? What do you mean by that?

The cats in the system are never sick. It is also vital to emphasize
I am talking about _many_ cats, not a single cat. Issues in quantum
theory cannot be understood by reflecting on a solitary cat. One
needs a great many of them, all identical. This is not a multiverse.
It is a system constructed out of lots of cats, half of them standing,
prancing, and playing, the other half stone cold dead. I wish to
collect all my data together of what I can expect to find, and that is
the superposition of live and dead cats, never sick, unambiguously
live and dead.

Doug

[Materion]

On Feb 23, 2:00pm, "Salviati" <eckard.blumsch...@arcor.de> wrote:
> Doug Sweetser did not yet get an answer to his question
>
> > Anyone recall the conjugate for number of photons?
> > It is the light intensity?

Isn't it the phase ?

k_x * x = phase
E * t / hbar = phase
(how does it work for spin?) = phase
....
N * phase = phase

One of the meanings of the Heisenberg relations is that the phase of a
quantum system has a total indeterminacy (2*pi).

Best regards,
Arjen Dijksman

[Hendrik Boom]

[Mod. note: Please keep replies on topic, i.e. relevant to physics. -ik ]

On Wed, 20 Feb 2008 09:37:35 +0000, Doug Sweetser wrote:

> Hello Salviati:
>
> I like this quote:
>
>> The reals are uncountable.
>
> That had deep implications for our description of Nature. What those
> implications are can be confusing.

But our measurements are rationals. The rationals are countable. This
too has deep implications for our description of Nature. The reals are
formed by some sort of completion of the rationals (Cauchy sequences or
Dedekind cuts; I don't care). The idea that such completion is possible
assumes we can have arbitrarily small intervals of rational numbers. If
we're talking about Nature, this would involve arbitrarily precise
measurements, which we know do not exist.

The closest we get to real numbers is limits of averages of ever-
increasing numbers of measurements. But we can't make infinite numbers
of measurements ...

-- hendrik

[Gerry Quinn]

In article <11ab6$47c5a5f0$d88ac3c2$3834@PRIMUS.CA>,
hendrik@topoi.pooq.com says...
> [Mod. note: Please keep replies on topic, i.e. relevant to physics. -ik ]
> On Wed, 20 Feb 2008 09:37:35 +0000, Doug Sweetser wrote:
> > Hello Salviati:
> >
> >> The reals are uncountable.
> >
> > That had deep implications for our description of Nature. What those
> > implications are can be confusing.
>
> But our measurements are rationals. The rationals are countable. This
> too has deep implications for our description of Nature. The reals are
> formed by some sort of completion of the rationals (Cauchy sequences or
> Dedekind cuts; I don't care). The idea that such completion is possible
> assumes we can have arbitrarily small intervals of rational numbers. If
> we're talking about Nature, this would involve arbitrarily precise
> measurements, which we know do not exist.
>
> The closest we get to real numbers is limits of averages of ever-
> increasing numbers of measurements. But we can't make infinite numbers
> of measurements ...

I don't see that our measurements are necessarily rationals. If we
observe that something has rotated exactly once, we can also say it has
rotated through 2*pi radians, an irrational number. If we measure a
value by determining its square, the quantity itself will be a square
root, which is typically irrational.

What is correct, I think, is the second point - we cannot make infinite
numbers of measurements. That means that our description of any
physical quantity - whether it uses rationals, irrationals, or any other
sort of number - must be finite. 2*pi is written as an infinitely long
non-recurring decimal, but it can be expressed quite briefly in various
ways.

Does that mean that the quantities *themselves* (as distinct from our
descriptions) must be finitely describable? That brings us back on
topic... one possible answer, it seems to me, is that classical objects,
i.e. objects entangled with an effectively infinite environment, may
have a state that cannot be finitely described.

- Gerry Quinn

[Gerry Quinn]

In article <c15007a2-346c-4b96-a0ed-
5678332bb59d@p73g2000hsd.googlegroups.com>, dougsweetser@gmail.com
says...
> Hello Gerry:
> You wrote:
>
> > A particular atom, say the one creating the top serif
> > of the 'D', can consistently be observed to be in the same place, and it
> > is not moving anywhere. There is a real sense in which its position is
> > known exactly, and its momentum is zero. It is living in the classical
> > world, precisely because of its entanglement with the other atoms.
>
> One certainly can do a series of experiments where the atoms are
> proven by observation to be sitting right on top of the 'D'. One
> could do another series of experiments to show that the momentum is
> zero. One cannot do a single experiment to show that in a measurable
> sense, "its position is known exactly, and its momentum is zero"
> without violating the uncertainty principle, the variation of the
> measurement of the position x times the variation in the measurement
> of the momentum px must be greater than the super tiny hbar.

Indeed. But why would you need to do any measurements when you know
exactly where it is at all times? For many purposes, the uncertainty
principle is irrelevant when it comes to this atom. As I was saying,
for these purposes it's living in the classical world.

> This was an instructive comment:
>
> > The atoms in the solid are strongly entangled with others
> > in their environment - the electrons in your vaguely-described 'cloud',
> > presumably, are not.
>
> The cloud is both very precise and reproducible. There are many
> systems that with their superposition of states have a location where
> there is zero probability to find the atom. An experiment can be set
> up to measure the probability distribution of the system over space,
> and we find that there are places with zero probability. We label
> this quantum interference. Unfortunately, we bring with us the notion
> of classical interference, where one thing gets together with another
> thing destructively. That is not the way quantum systems work -
> everything is independent.

I'm not sure what you are getting at here. Our usual model of quantum
interference is based on the same equations as the interference of
idealised classical waves. (The interactions of classical waves such as
ripples in water are never completely linear, of course.) In this model
the positions where the probability of finding an atom is zero are those
where the wave functions associated with possible states/transitions of
the system interfere destructively.

> > You could semi-plausibly have a superposition state that looked like a
> > cat that you could not tell was alive or dead. It would instantly
> > collapse into one or the other, or more likely into a very sick cat.
> ..
> > So you are taking the classical results observed after opening each box,
> > and superimposing them. Just what I said. No picture of an observed
> > superposition state. No indication of interference between dead and
> > alive states.
>
> This indicates we are not communicating so well on these issues, which
> is not uncommon. There is no need for a sick cat state. Nor is there
> a need to show interference between the dead and alive states. As
> soon as I discuss making a measurement of anything, you slap the label
> "classical result", a behavior I find mystifying. The CCD camera at
> the end of a two slit interference experiment would appear to fit this
> notion of classical result since the signals are either on or off.

Yes - the photograph of an interference pattern in a two-slit experiment
is entirely classical. But the picture shows an interference pattern -
something that is not present in your picture of Schrodinger's cat. The
latter picture could be drawn by somebody who has never heard of quantum
mechanics.

> So here is your definition:
>
> > Classical result = what you get when you open the box and see a dead cat
> > or a live cat. Nothing more.
>
> Unfortunately, I don't get what you mean. For me, what classical
> physics is about is our ability to watch a system, say a live cat,
> evolve in time. We can watch a cat go from alive to sick to dead over
> a period of observation.

But the period required for decoherence in a cat is so short that the
smallest instant you could watch it for is more than long enough!

> The SSCE I described ("Simplified Schrodinger's Cat Experiment"),

Hey, that was my phrase, describing a different experiment :-)

> there is no observation you can ever do with the cat transitioning in
> time from live to dead. One gets one or the other. The best we can
> do to summarize the results is a picture like I provided.

Which means that there is no quantum mechanics to see in the summary of
the results.

> Neat, I see a connection to quantum interference. Earlier I had said
> there is no "sick cat" state. That is a state the classical physicist
> would expect to see, part of the transition in time from live to
> dead. In the precise, repeatable system I set up with a thousand cat
> clones, not a single one was sick. It is the omission of expected
> states that many find troubling about quantum mechanics.

Again, this omission is purely classical. If you repeat the experiment
enough times, the proportion of sick cats will be the (approximately)
same as the proportion of (non-poisoned) cats you would classically have
expected to fall ill. There is no constructive or destructive
interference - just classically additive probabilities.

The transition from live to dead in the case of the cats for whom the
poison bottle was broken will be observed to have occurred before the
box was opened, in all of the boxes where the bottle was broken. Traces
of it will be detectable if the cat is autopsied.

> > "Unambiguously live and dead"? What do you mean by that?
>
> The cats in the system are never sick. It is also vital to emphasize
> I am talking about _many_ cats, not a single cat. Issues in quantum
> theory cannot be understood by reflecting on a solitary cat. One
> needs a great many of them, all identical. This is not a multiverse.
> It is a system constructed out of lots of cats, half of them standing,
> prancing, and playing, the other half stone cold dead.

But none exhibiting any real quantum mechanical behaviour.

> I wish to
> collect all my data together of what I can expect to find, and that is
> the superposition of live and dead cats, never sick, unambiguously
> live and dead.

As I pointed out above, the sick state (a transition state between live
and dead) will be observable via autopsy or perhaps other signs in all
of the cases where the cat is found dead. It will be observed to have
occurred before the box was opened.

None of this is in any way surprising, given that the purpose of
Scrodinger's thought experiment was to point out the oddness of quantum
theory, and ask how we should interpret it - not to demonstrate its
truth. For the latter, the two-slit experiment, experiments with
entangled photon pairs, etc., are best. A nineteenth-century physicist
would have correctly predicted the results that would be obtained from
the Schrodinger Cat experiment, because the predictions of quantum
theory and classical theory are the same for it.

- Gerry Quinn

[nanobug]

>
>I am trying to find an objective definition of 'measurement', one
>that need not involve physicists, other humans, or cats.
>

You might want to have a look at the concept (and process) of decoherence:

http://www.ipod.org.uk/reality/reality_decoherence.asp

[Hendrik Boom]

On Sat, 01 Mar 2008 16:38:32 +0000, Gerry Quinn wrote:

> In article <11ab6$47c5a5f0$d88ac3c2$3834@PRIMUS.CA>,
> hendrik@topoi.pooq.com says...
>> [Mod. note: Please keep replies on topic, i.e. relevant to physics. -ik ]
>> On Wed, 20 Feb 2008 09:37:35 +0000, Doug Sweetser wrote:
>> > Hello Salviati:
>> >
>> >> The reals are uncountable.
>> >
>> > That had deep implications for our description of Nature. What those
>> > implications are can be confusing.
>>
>> But our measurements are rationals. The rationals are countable. This
>> too has deep implications for our description of Nature. The reals are
>> formed by some sort of completion of the rationals (Cauchy sequences or
>> Dedekind cuts; I don't care). The idea that such completion is possible
>> assumes we can have arbitrarily small intervals of rational numbers. If
>> we're talking about Nature, this would involve arbitrarily precise
>> measurements, which we know do not exist.
>>
>> The closest we get to real numbers is limits of averages of ever-
>> increasing numbers of measurements. But we can't make infinite numbers
>> of measurements ...
>
> I don't see that our measurements are necessarily rationals. If we
> observe that something has rotated exactly once, we can also say it has
> rotated through 2*pi radians, an irrational number. If we measure a
> value by determining its square, the quantity itself will be a square
> root, which is typically irrational.
>
> What is correct, I think, is the second point - we cannot make infinite
> numbers of measurements. That means that our description of any
> physical quantity - whether it uses rationals, irrationals, or any other
> sort of number - must be finite. 2*pi is written as an infinitely long
> non-recurring decimal, but it can be expressed quite briefly in various
> ways.
>
> Does that mean that the quantities *themselves* (as distinct from our
> descriptions) must be finitely describable? That brings us back on
> topic... one possible answer, it seems to me, is that classical objects,
> i.e. objects entangled with an effectively infinite environment, may
> have a state that cannot be finitely described.

Our measurements are finitely describable. Whether the actual quantities
are finitely describable is not something we can determine experimentally.
Such infinitude could of course be a nonobservable part of a theory.

-- hendrik


>
> - Gerry Quinn