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Sep 1, 2022, 12:30:58â€¯PM9/1/22

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Can anyone give a clear explanation why information has to be conserved

in quantum mechanics?

I was not taught this when first learning about QM in the 1970's. As

best as I can tell the idea comes from the idea that the QM wave

function evolves per a unitary operator that can, in principle, be

reversed to recover the past state as well as calculate the future state

of the system.

It seems to me that this argument is missing two important facts: -The

wave function is not real, it is only a mathematical tool for predicting

the probabilities of future states -The actual future is one of many

predicted by the wave function, and likewise can be the result of many

different possible past states.

It seems to me that each time the wave function "collapses" that

information is lost. Is there a good argument why this is wrong?

Rich L.

in quantum mechanics?

I was not taught this when first learning about QM in the 1970's. As

best as I can tell the idea comes from the idea that the QM wave

function evolves per a unitary operator that can, in principle, be

reversed to recover the past state as well as calculate the future state

of the system.

It seems to me that this argument is missing two important facts: -The

wave function is not real, it is only a mathematical tool for predicting

the probabilities of future states -The actual future is one of many

predicted by the wave function, and likewise can be the result of many

different possible past states.

It seems to me that each time the wave function "collapses" that

information is lost. Is there a good argument why this is wrong?

Rich L.

Sep 3, 2022, 2:34:47â€¯AM9/3/22

to

On Thursday, 1 September 2022 at 18:30:58 UTC+2, richali...@gmail.com wrote:

>

> Can anyone give a clear explanation why information has to be conserved

> in quantum mechanics?

>

> I was not taught this when first learning about QM in the 1970's. As

> best as I can tell the idea comes from the idea that the QM wave

> function evolves per a unitary operator that can, in principle, be

> reversed to recover the past state as well as calculate the future

> state of the system.

That information is conserved follows from unitarity which in turn
>

> Can anyone give a clear explanation why information has to be conserved

> in quantum mechanics?

>

> I was not taught this when first learning about QM in the 1970's. As

> best as I can tell the idea comes from the idea that the QM wave

> function evolves per a unitary operator that can, in principle, be

> reversed to recover the past state as well as calculate the future

> state of the system.

follows from reversibility, i.e. that the laws of physics are the same

whether we let the system evolve forwards or backwards in time (indeed,

except for the "measurement problem" as well as thermodynamics).

And *reversibility* itself is already present in classical physics,

namely since the Hamiltonian/Lagrangian approaches, where the dynamics

of a system are described in terms of the evolution of a *system state*

in a state space: *that* evolution has to be reversible, indeed lack of

reversibility would simply not be a state space: IOW, given a "law of

motion", there must be one and only one next state, and one and only one

previous state, or the whole state-space based approach becomes simply

meaningless.

> It seems to me that this argument is missing two important facts: -The

> wave function is not real, it is only a mathematical tool for predicting

> the probabilities of future states -The actual future is one of many

> predicted by the wave function, and likewise can be the result of many

> different possible past states.

issue of ontology: e.g. in pilot-wave theory the wave function *is* real.

And I'd personally agree that physics without a *solid* ontology is very

poor thing.

> It seems to me that each time the wave function "collapses" that

> information is lost. Is there a good argument why this is wrong?

"shut up and calculate". Is(n't) that wrong? Many think that it in

fact means an open problem, just there is sort of a taboo against

wave-particle duality (as in pilot-wave theory), despite it *is*

ontologically solid, and we are rather offered multiverses...

Julio

Sep 3, 2022, 9:50:03â€¯AM9/3/22

to

Julio Di Egidio <ju...@diegidio.name> schrieb:

> And *reversibility* itself is already present in classical physics,

> namely since the Hamiltonian/Lagrangian approaches, where the dynamics

> of a system are described in terms of the evolution of a *system state*

> in a state space: *that* evolution has to be reversible, indeed lack of

> reversibility would simply not be a state space: IOW, given a "law of

> motion", there must be one and only one next state, and one and only one

> previous state, or the whole state-space based approach becomes simply

> meaningless.

And yet, the Second Law of Thermodynamics holds.

> And *reversibility* itself is already present in classical physics,

> namely since the Hamiltonian/Lagrangian approaches, where the dynamics

> of a system are described in terms of the evolution of a *system state*

> in a state space: *that* evolution has to be reversible, indeed lack of

> reversibility would simply not be a state space: IOW, given a "law of

> motion", there must be one and only one next state, and one and only one

> previous state, or the whole state-space based approach becomes simply

> meaningless.

Sep 3, 2022, 8:15:37â€¯PM9/3/22

to

Thermodynamics is indeed another story: the laws of thermodynamics

are not exact laws, in fact thermodynamics is not deterministic, while,

to the point, the laws of quantum mechanics *are* exact and

deterministic: the fact that we measure probabilities has again to do

with "the problem of measurement", i.e. how we go from the quantum

state to a classical outcome, but, to reiterate, the evolution of the wave

function, as expressed by the Schroedinger equation, is per se indeed

deterministic.

Julio

Sep 4, 2022, 1:54:28â€¯PM9/4/22

to

Thank you for your detailed response. I have to debate some points where

I think you are being inconsistent. You say that QM wave function is

reversible because it evolves by a unitary operator. This is true, but the

wave function calculated is not real. I admit that it is simulating something

that is real, as proven by interference patterns, but that is not the same thing.

The difference is "collapse". Take the two slit experiment with a single

photon. When a photon is detected on the screen at one of several

predicted interference spots, the physical situation is now a photon

(or its packet of energy-momentum) in one specific atom/molecule.

If you try to start from this state and compute backwards you do NOT

get that the photon had to come from the single point light source on

the other side of the screen. The backward computed wave function

includes many possible sources, including the one that photon

actually came from.

QM is not deterministic in that at any given moment

there are several possible mutually exclusive future outcomes. It seems

to me that this means that information is not conserved in the actual

physics, despite what the math appears to say. The disconnect is that

the math only predicts probabilities of multiple outcomes, while the

actual physics, in many but not all cases, has only one specific future.

In other words, knowing the exact present state does not tell you which

of many future states the system may evolve to, nor which of several

states it evolved from. It seems to me that this precludes any

conservation of information.

Is there an error in my reasoning here?

Rich L.

Sep 5, 2022, 1:03:11â€¯AM9/5/22

to

On Sunday, 4 September 2022 at 19:54:28 UTC+2, richali...@gmail.com wrote:

> On Saturday, September 3, 2022 at 7:15:37 PM UTC-5, ju...@diegidio.name wrote:

> > On Saturday, 3 September 2022 at 15:50:03 UTC+2, Thomas Koenig wrote:

> > > Julio Di Egidio <ju...@diegidio.name> schrieb:

> > > >

> > > > And *reversibility* itself is already present in classical physics,

<snip>
> On Saturday, September 3, 2022 at 7:15:37 PM UTC-5, ju...@diegidio.name wrote:

> > On Saturday, 3 September 2022 at 15:50:03 UTC+2, Thomas Koenig wrote:

> > > Julio Di Egidio <ju...@diegidio.name> schrieb:

> > > >

> > > > And *reversibility* itself is already present in classical physics,

> Thank you for your detailed response. I have to debate some points where

> I think you are being inconsistent.

Even if you disagree or don't see the point, that doesn't make
> I think you are being inconsistent.

what I said (which is anyway just basic QM) inconsistent. In

fact, below you are repeating exactly the same mistakes as

in your opening post. So, I'll just repeat quickly:

<snipped>

> but the wave function calculated is not real.

That *depends* on your ontological stance: and your choice,
which is the standard one, is the one that is *most* problematic.

> The difference is "collapse". Take the two slit experiment with

> If you try to start from this state and compute backwards [...]

> The backward computed wave function includes many

> possible sources

When the photon is detected, that is a *measurement*, i.e.
> possible sources

(standardly) you have *collapsed* the wave function, aka

the state: and *that* operation is not reversible, not QM!!

Conversely, do not collapse the wave function (consider

the joint system observer/observed) and you stay quantum

and have reversibility.

> QM is not deterministic in that at any given moment there

> are several possible mutually exclusive future outcomes.

the evolution of the wave function, aka the quantum state.

> to me that this means that information is not conserved

> Is there an error in my reasoning here?

HTH,

Julio

Sep 5, 2022, 1:05:34â€¯AM9/5/22

to

On Sunday, September 4, 2022 at 10:54:28 AM UTC-7, richali...@gmail.com wro=

[[Mod. note -- Quoted text trimmed. -- J.T.]]

> QM is not deterministic in that at any given moment

> there are several possible mutually exclusive future outcomes. It seems

> to me that this means that information is not conserved in the actual

> physics, despite what the math appears to say. The disconnect is that

> the math only predicts probabilities of multiple outcomes, while the

> actual physics, in many but not all cases, has only one specific future.

>

> In other words, knowing the exact present state does not tell you which

> of many future states the system may evolve to, nor which of several

> states it evolved from. It seems to me that this precludes any
> to me that this means that information is not conserved in the actual

> physics, despite what the math appears to say. The disconnect is that

> the math only predicts probabilities of multiple outcomes, while the

> actual physics, in many but not all cases, has only one specific future.

>

> In other words, knowing the exact present state does not tell you which

> of many future states the system may evolve to, nor which of several

> conservation of information.

>

> Is there an error in my reasoning here?

>

> Rich L.
> Is there an error in my reasoning here?

>

A naive query: I've known qualitatively that unitarity guarantees

that no information is ever destroyed. Does unitarity allow for

the creation of information? My sense is that just as our physical

universe evolves in a direction of greater complexity, this also

means toward greater amounts of information.

Sep 7, 2022, 7:35:50â€¯AM9/7/22

to

On Monday, September 5, 2022 at 12:03:11 AM UTC-5, ju...@diegidio.name wrote:

> On Sunday, 4 September 2022 at 19:54:28 UTC+2, richali...@gmail.com wrote:

> > On Saturday, September 3, 2022 at 7:15:37 PM UTC-5, ju...@diegidio.name wrote:

> <snipped>

> > but the wave function calculated is not real.

> That *depends* on your ontological stance: and your choice,

> which is the standard one, is the one that is *most* problematic.

This clarifies a difference of opinion between us. I don't understand
> On Sunday, 4 September 2022 at 19:54:28 UTC+2, richali...@gmail.com wrote:

> > On Saturday, September 3, 2022 at 7:15:37 PM UTC-5, ju...@diegidio.name wrote:

> <snipped>

> > but the wave function calculated is not real.

> That *depends* on your ontological stance: and your choice,

> which is the standard one, is the one that is *most* problematic.

why you say that is *most* problematic? Can you elaborate? I'm

curious.

> > The difference is "collapse". Take the two slit experiment with

> > a single photon. When a photon is detected on the screen [...]

> > If you try to start from this state and compute backwards [...]

> > The backward computed wave function includes many

> > possible sources

> When the photon is detected, that is a *measurement*, i.e.

> (standardly) you have *collapsed* the wave function, aka

> the state: and *that* operation is not reversible, not QM!!

> Conversely, do not collapse the wave function (consider

> the joint system observer/observed) and you stay quantum

> and have reversibility.

physics? Aren't there some events that permanently refine the wave

function down to a subset of possible outcomes?

For what it is worth, I distinguish the physics of QM from the math of

QM. The math is a model of our understanding that may be imperfect. I

believe there is a reality where the actual physics happens.

> > QM is not deterministic in that at any given moment there

> > are several possible mutually exclusive future outcomes.

> Yes, it is: you are still conflating classical outcomes with

> the evolution of the wave function, aka the quantum state.

> > to me that this means that information is not conserved

> Sure, but it's *collapse* that destroys information.

> > Is there an error in my reasoning here?

> Only if you'll insist... :)

Seriously, isn't collapse a part of reality, or required by reality?

For example, back to the photon detected in the two slit experiment,

once the photon is detected there is now a new reality that differs from

the previous wave function.

It sounds like you may follow the many worlds concept. If so I can see

your point being that the wave function describes all the possible

worlds and then no information would be lost. However we live in just

one of those many worlds, and it seems to me that we need a QM theory

that includes the random selection (aka "measurement" or "collapse")

down to just one of the many worlds. Therefore it seems to me that in

our world there is a loss of information every time there is a

measurement.

Or so I believe!

Thanks Julio for responding, this is clarifying these ideas for me.

Rich L.

>

> HTH,

>

> Julio

Sep 8, 2022, 5:13:27â€¯AM9/8/22

to

On 9/7/22 6:35 AM, Richard Livingston wrote:

> [...] Seriously, isn't collapse a part of reality, or required by

> reality?

No. There are interpretations of QM that do not involve any "collapse of

the wavefunction". See, for example:

Ballentine, _Quantum_Mechanics:_A_Modern_Development_.

The basic idea is that whenever one makes a measurement of a quantum

system, that necessarily involves coupling it to a MUCH LARGER measuring

instrument, and the comparatively tiny quantum system is "forced" into

an appropriate eigenstate by that coupling.

Tom Roberts

> [...] Seriously, isn't collapse a part of reality, or required by

> reality?

No. There are interpretations of QM that do not involve any "collapse of

the wavefunction". See, for example:

Ballentine, _Quantum_Mechanics:_A_Modern_Development_.

The basic idea is that whenever one makes a measurement of a quantum

system, that necessarily involves coupling it to a MUCH LARGER measuring

instrument, and the comparatively tiny quantum system is "forced" into

an appropriate eigenstate by that coupling.

Tom Roberts

Sep 8, 2022, 5:13:27â€¯AM9/8/22

to

On Wednesday, 7 September 2022 at 13:35:50 UTC+2, richali...@gmail.com wrote:

> On Monday, September 5, 2022 at 12:03:11 AM UTC-5, ju...@diegidio.name wrote:

> > On Sunday, 4 September 2022 at 19:54:28 UTC+2, richali...@gmail.com wrote:

> > > On Saturday, September 3, 2022 at 7:15:37 PM UTC-5, ju...@diegidio.name wrote:

<snipped>

> > > but the wave function calculated is not real.

>

> > That *depends* on your ontological stance: and your choice,

> > which is the standard one, is the one that is *most* problematic.

>

> This clarifies a difference of opinion between us. I don't understand

> why you say that is *most* problematic? Can you elaborate? I'm

> curious.

For the reasons I have hinted at upthread: 1) saying "the wave function
> On Monday, September 5, 2022 at 12:03:11 AM UTC-5, ju...@diegidio.name wrote:

> > On Sunday, 4 September 2022 at 19:54:28 UTC+2, richali...@gmail.com wrote:

> > > On Saturday, September 3, 2022 at 7:15:37 PM UTC-5, ju...@diegidio.name wrote:

<snipped>

> > > but the wave function calculated is not real.

>

> > That *depends* on your ontological stance: and your choice,

> > which is the standard one, is the one that is *most* problematic.

>

> This clarifies a difference of opinion between us. I don't understand

> why you say that is *most* problematic? Can you elaborate? I'm

> curious.

is not real" means a plain hole in a theory's ontology (you may think our

physical theories are full of these "holes": they aren't, and the problem

here is in fact quite serious), and in this respect the standard interpretation

is the weakest of the lot, i.e. as compared to pilot-wave theory or even

many-worlds ("multiverses"); 2) "collapse" of the wave function is indeed

an externally added postulate that simply has no place in any physical

theory proper (under the least-action over a state space paradigm that is);

and, 3) with collapse we break "reversibility", which is really the nail in the

coffin as far as that theory is concerned. So... "shut up and calculate."

Indeed, note that many-worlds was born exactly as an attempt to solve

the problems of the standard interpretation: there is no "collapse" but

rather "branching" in many-worlds, and, while ontologically that remains

quite unjustified (as long as there is no way to probe the existence of

these branches), it at least saves reversibility.

While, as for pilot wave theory, that is fine ontologically and otherwise,

indeed (as I get it) it is the best quantum theory we could have, but it's

simply been "cancelled". Sure, it's dualistic, just as in Yin and Yang...

but this is another story.

> For what it is worth, I distinguish the physics of QM from the math of

> QM. The math is a model of our understanding that may be imperfect. I

> believe there is a reality where the actual physics happens.

dimensions, theoretical and applicative (experimental), where the

theoretical part is where you find the (physical!) models: and the more

we probe into realms that we cannot directly experience, the more the

theoretical part becomes relevant and needs to be solid. As for maths,

that is simply a tool, it provides a formal language and algebra, but that's

all about it, it certainly does not dictate anything properly physical.

That said, the belief you express above I agree with (how couldn't I), but

now you are getting into the properly philosophical issue of what one

believes about cosmos: physics per se just needs a solid ontology and

then it is proper physics, whether or not the whole cosmos is somebody's

dream or else... but this too is another story and only marginal to our topic.

> Seriously, isn't collapse a part of reality, or required by reality?

theory* to somehow manage to use it at all. Conversely, the fact that

we "measure things" does not per se entail the need for "collapse".

> For example, back to the photon detected in the two slit experiment,

> once the photon is detected there is now a new reality that differs from

> the previous wave function.

thread: at this point I think if you want more or even just more confirmation,

you should rather start looking into some course material...

> It sounds like you may follow the many worlds concept.

> Thanks Julio for responding, this is clarifying these ideas for me.

Julio

Sep 8, 2022, 6:37:10â€¯PM9/8/22

to

On Thursday, September 8, 2022 at 4:13:27 AM UTC-5, Tom Roberts wrote:

> On 9/7/22 6:35 AM, Richard Livingston wrote:

> > [...] Seriously, isn't collapse a part of reality, or required by
> On 9/7/22 6:35 AM, Richard Livingston wrote:

> > reality?

>

> No. There are interpretations of QM that do not involve any "collapse of

> the wavefunction". See, for example:

>

> Ballentine, _Quantum_Mechanics:_A_Modern_Development_.

>

> The basic idea is that whenever one makes a measurement of a quantum

> system, that necessarily involves coupling it to a MUCH LARGER measuring

> instrument, and the comparatively tiny quantum system is "forced" into

> an appropriate eigenstate by that coupling.

>

> Tom Roberts

I'm having a real problem with this idea. It seems to me that there
> the wavefunction". See, for example:

>

> Ballentine, _Quantum_Mechanics:_A_Modern_Development_.

>

> The basic idea is that whenever one makes a measurement of a quantum

> system, that necessarily involves coupling it to a MUCH LARGER measuring

> instrument, and the comparatively tiny quantum system is "forced" into

> an appropriate eigenstate by that coupling.

>

> Tom Roberts

are certain events that clearly result in the wave function changing

radically. I gave one example with a photon detected in one particular

spot on a screen. Another would be a isolated atom that emits a photon.

The QM treatment gives a wave function that expands outward in all

directions. Eventually (perhaps years later) that photon is absorbed

by some distant atom. Energy has been transferred from one location to

another at a later time. This is a very real event and experimentally

verifiable.

If it had been scattered immediately with negligible loss of energy, I

could see that as a possible evolution of the state without any collapse

or loss of information. However if it is absorbed and eventually

converted into heat, isn't that equivalent to collapse of the wave

function? Isn't that irreversible? Doesn't that constitute loss of

information?

I understand that some "interpretations" of QM, such as many worlds,

avoid this by partitioning the information into multiple adjacent but

inaccessible adjacent worlds, but it seems to me that this is an

untestable theory, and therefore unscientific.

I have Ballantines book, I'll go back into it to see what he says about

all this.

Rich L.

Sep 10, 2022, 4:51:48â€¯AM9/10/22

to

On 9/8/22 4:13 AM, Julio Di Egidio wrote:

> [...] saying "the wave function is not real" means a plain hole in a

> theory's ontology

Say "model" instead of "theory", and you'll more easily see the basic

mistake you are making here: no human thought has any ontology at all,

because thoughts do not exist in the world we inhabit, they "exist" in

our minds only via a PUN on "exist". This of course applies to our

physical models of the world, such as QM: they are pure thoughts.

This has been a fundamental evolution in physics: we are NOT "describing

how the world works", we are MODELING how we observe the world to work.

So, for instance, there is no expectation that quantities that appear in

a model necessarily correspond to some object or process in the world.

The difference is profound, and involves a radical change in outlook and

approach. Remember that "model" inherently implies validity only in a

limited domain....

[So, for instance, the incompatibility between GR and QM

does not imply a fundamental problem, as they are both

excellent models valid in disjoint domains. We are

searching for a better model that will be valid in a

wider domain that hopefully encompasses both of their

individual domains.]

Tom Roberts

> [...] saying "the wave function is not real" means a plain hole in a

> theory's ontology

Say "model" instead of "theory", and you'll more easily see the basic

mistake you are making here: no human thought has any ontology at all,

because thoughts do not exist in the world we inhabit, they "exist" in

our minds only via a PUN on "exist". This of course applies to our

physical models of the world, such as QM: they are pure thoughts.

This has been a fundamental evolution in physics: we are NOT "describing

how the world works", we are MODELING how we observe the world to work.

So, for instance, there is no expectation that quantities that appear in

a model necessarily correspond to some object or process in the world.

The difference is profound, and involves a radical change in outlook and

approach. Remember that "model" inherently implies validity only in a

limited domain....

[So, for instance, the incompatibility between GR and QM

does not imply a fundamental problem, as they are both

excellent models valid in disjoint domains. We are

searching for a better model that will be valid in a

wider domain that hopefully encompasses both of their

individual domains.]

Tom Roberts

Sep 10, 2022, 5:39:25â€¯AM9/10/22

to

On Saturday, 10 September 2022 at 10:51:48 UTC+2, Tom Roberts wrote:

> On 9/8/22 4:13 AM, Julio Di Egidio wrote:

> > [...] saying "the wave function is not real" means a plain hole in a

> > theory's ontology

>

> Say "model" instead of "theory", and you'll more easily see the basic

> mistake you are making here:

No, you are simply way out of your depths.
> On 9/8/22 4:13 AM, Julio Di Egidio wrote:

> > [...] saying "the wave function is not real" means a plain hole in a

> > theory's ontology

>

> Say "model" instead of "theory", and you'll more easily see the basic

> mistake you are making here:

> no human thought has any ontology at all,

> because thoughts do not exist in the world we inhabit, they "exist" in

of "ontology", or of "theory" for that sake.

> This has been a fundamental evolution in physics: we are NOT "describing

> how the world works", we are MODELING how we observe the world to work.

mathematical or else) theory needs an ontology to be

a theory at all: with physics, the meaning and value of

models indeed is exactly proportional to their correspondence

to the "object of study", aka, from a theoretical point of view,

the ontology.

> So, for instance, there is no expectation that quantities that appear in

> a model necessarily correspond to some object or process in the world.

You must have implicitly internalized the shut up and calculate,

as otherwise that is in no way how physics or any other science

is supposed to work.

And now a word a caution to you specifically, Tom, since you

regularly seem to jump on what I write: while I am relatively new

to physics, I happen to be an expert in foundations as well as,

FWIW, in philosophy generally. So, please, rather ask questions.

HTH and EOD (hopefully).

Julio

Oct 7, 2022, 4:17:28â€¯AM10/7/22

to

On Thursday, September 1, 2022 at 11:30:58 AM UTC-5, rich... wrote:

> It seems to me that this argument is missing two important facts: -The

> wave function is not real, it is only a mathematical tool for predicting

> the probabilities of future states

There is no such thing as being "only a mathematical tool".
> It seems to me that this argument is missing two important facts: -The

> wave function is not real, it is only a mathematical tool for predicting

> the probabilities of future states

Everything that says anything meaningful about the physical world

is ipso facto physical and physically relevant, and is thus real.

Anything that has any effect or impact on the physical world

is thereby physical; even thoughts and things called "subjective"

(since the brain, itself, is a physical object).

Physicality is the Borg of all attributes.

It assimilates everything that comes into contact with it.

Oct 7, 2022, 12:08:44â€¯PM10/7/22

to

On Friday, October 7, 2022 at 3:17:28 AM UTC-5, Rock Brentwood wrote:

> On Thursday, September 1, 2022 at 11:30:58 AM UTC-5, rich... wrote:

> > It seems to me that this argument is missing two important facts: -The

> > wave function is not real, it is only a mathematical tool for predicting

> > the probabilities of future states

> There is no such thing as being "only a mathematical tool".

> Everything that says anything meaningful about the physical world

> is ipso facto physical and physically relevant, and is thus real.

...
> On Thursday, September 1, 2022 at 11:30:58 AM UTC-5, rich... wrote:

> > It seems to me that this argument is missing two important facts: -The

> > wave function is not real, it is only a mathematical tool for predicting

> > the probabilities of future states

> There is no such thing as being "only a mathematical tool".

> Everything that says anything meaningful about the physical world

> is ipso facto physical and physically relevant, and is thus real.

I have to clarify my statement on the wave function. It is both a

mathematical tool and also something that somehow, probably imperfectly,

mirrors something real. To take the wave function as something real is,

I think, not justified by the non-physical behaviors, e.g. the

"instantaneous collapse". Never the less, there is something real that

the wave function is representing, and I think more of us should be

trying to figure out what that is.

Rich L.

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