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Author Topic:   A definition of infinity?
Casey Powell 
Inactive Suspended Member


Message 31 of 41 (374115)
01-03-2007 6:03 PM
Reply to: Message 27 by JavaMan
12-28-2006 7:39 AM


Re: Forever and ever and ever
Yup....how do we imagine an infinite being such as God for instance?
This message is a reply to:
 Message 27 by JavaMan, posted 12-28-2006 7:39 AM JavaMan has responded

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 Message 32 by JavaMan, posted 01-04-2007 8:10 AM Casey Powell has not yet responded

  
JavaMan
Member (Idle past 482 days)
Posts: 475
From: York, England
Joined: 08-05-2005


Message 32 of 41 (374333)
01-04-2007 8:10 AM
Reply to: Message 31 by Casey Powell
01-03-2007 6:03 PM


Re: Forever and ever and ever
Yup....how do we imagine an infinite being such as God for instance?

What is it you imagine?

Personally, my notion of God is rather vague. It's just what I've picked up from reading, and from Sunday School classes as a child. As I don't believe there is such a thing, I haven't explored my notion any further.


'I can't even fit all my wife's clothes into a suitcase for travelling. So you want me to believe we're going to put all of the planets and stars and everything into a sandwich bag?' - q3psycho on the Big Bang
This message is a reply to:
 Message 31 by Casey Powell, posted 01-03-2007 6:03 PM Casey Powell has not yet responded

  
Son Goku
Member
Posts: 1121
From: Ireland
Joined: 07-16-2005


Message 33 of 41 (374436)
01-04-2007 2:25 PM
Reply to: Message 26 by dogrelata
12-28-2006 6:19 AM


Measurement, paths and Zeno freezing.
Sorry for the delay, I was a bit busy.
I also apologize in advance if I ramble a bit and repeat things.

I'll expand on my message (No. 28) and answer Javaman's question first.

Javaman writes:

Could you explain this a little further? Classical probability is just a statistical thing: if path X has a probability of 0.2, all this means is that in 100 cases of the ball travelling from A to B, then the ball is likely to follow path X in 20 of them. What does the probability of 0.2 mean in the quantum case?


It means the exact same thing in the quantum case.
Take a look at the picture above and consider the circumstance where two of the paths are absent, leaving only one, let's say the top one.

In both the classical and quantum case, if I send a ball down the path it has a probability of 33.3% of reaching B. Simple enough.

Now consider the case where only the middle path is present, in both the classical and quantum case, if I send a ball down that path I've a 33.3%........e.t.c.

Now consider the scenario with only the bottom path.......e.t.c.

So far no difference between classical and quantum cases.
Now what happens when I consider the case where all three paths are present?

In the classical case I start with one path and open up the other two. The process being:
Starting probability: 33.3%
Open one path: +33.3%
Open the next path: +33.3%
Total: 100.0%

In the quantum case:
Starting probability: 33.3%
Open one path: +33.3%
Open the next path: -66.6% (Even though it was 33.3% when open on its own)
Total: 0.0%

What happened when I opened the final path?
Basically, in the classical case I don't have to take into account the presence of other paths when I open another one. It just adds to the total probability.

In the quantum case, opening a new path can cause interference with the other two. When the path is open by itself, this interference is absent so it gives a regular 33.3% probability. With the interference from the other paths it gives -66.6%.

Now the thing is, in no quantum mechanical situation will we ever end up with a negative probability over all. Even though some paths will give negative probabilities there will always be a greater or equal contribution from the non-negative ones.

If we were to try and get an overall negative probability, we'd have to shut off the positive probability paths, but this kills off some of interference that makes the negative probability paths have negative probability. So they turn into positive probability paths, ensuring the whole system always has a positive probability.





Anyway on to dogrelata's post:

dogrelata writes:

I can get my head around the Schrödinger’s cat thought experiment. As far as I understand it, there is no way of knowing whether the cat is alive or dead until the box is opened. Although I have seen this characterized in some places, as being the cat is simultaneously alive and dead, my standard worldview kicks in and dismisses this as purely hypothetical. A different analogy might be, a football match has taken place, but I am unaware of what the result is, so until I learn of the result, the outcome remains a matter of probability-based conjecture.


Since a cat has the unfortunate properties of being mostly classical, I will instead consider the case of an atom on its own which starts in a state of having a 50/50 chance of decaying or not decaying.

I'll write the quantum state of the atom in the following form:
(a,b)

Where:
a=current chance of being mesured as decayed.
b=current chance of being measured as not decayed.

So when we start off the atom is in the state (50,50).
I make a measurement to see if the atom decayed. Let's say I find the atom is not decayed, so the state right after measurement is:
(0,100)
Obviously, because I've measured it to be not decayed it has 100% of being not decayed when I measure it.

Then I leave my equipment and stop measuring the atom. Over time the probability returns to 50/50.
So over a time of about a tenth of a second, the following happens to the state of the particle:
(0,100) -> (10,90) -> (20,80) -> (30,70) -> (40,60) -> (50,50).




Now I can introduce the Zeno effect, probably one of the weirdest things in QM.

Now that the atom is back in the 50/50 state, I get a piece of equipment that emits one frequency of light, let's say a pure red laser.
When the atom is decayed it absorbs the red light, when it's not decayed the light passes straight through it. So I emit the light at the atom. It passes straight through, meaning the atom is not decayed and therefore is in the (0,100) state.

As before the atom will start to evolve back into the 50/50 state. However let's say I measure the atom again, really quickly, so quick that it's only gotten to (1,99).
Since it has a 99% chance of being measured undecayed, the odds are I will measure it as not decayed by seeing the light to pass through the atom, which returns it to (0,100).

If I keep measuring it really fast, so that it only gets to about (0.0001,99.9999) every time, the odds are I won't measure it as decayed for quite a while, because the odds are much greater to measure it as not decayed.
In this way I can freeze the atom in an undecayed state for quite a long time.

However this is weird when you think about it. The way I measure it as not decayed is if light passes through it, i.e. if the atom and the light don't interact.

So I've managed to stop an atom from decaying by not interacting with it, in other words, by doing nothing to it.

(And this effect has been demonstrated in a lab.)

Edited by Son Goku, : Minor spelling correction.


This message is a reply to:
 Message 26 by dogrelata, posted 12-28-2006 6:19 AM dogrelata has responded

Replies to this message:
 Message 34 by dogrelata, posted 01-06-2007 3:11 AM Son Goku has not yet responded
 Message 35 by cavediver, posted 01-06-2007 9:36 AM Son Goku has responded
 Message 38 by JavaMan, posted 01-08-2007 8:55 AM Son Goku has responded

  
dogrelata
Member (Idle past 3475 days)
Posts: 201
From: Scotland
Joined: 08-04-2006


Message 34 of 41 (374879)
01-06-2007 3:11 AM
Reply to: Message 33 by Son Goku
01-04-2007 2:25 PM


Re: Measurement, paths and Zeno freezing.
Cheers SG. Thanks for taking the time to try and explain these things. It is much appreciated.

Hopefully I am starting to get a better grasp of the some of the issues involved, but am still hampered by my ‘standard worldview’ in the way I see things.

For example, with classical statistics, I can assume the chances of throwing a 6 when I roll a dice are 16.67%. To prove this hypothesis, all I need to do is roll the dice a sufficient number of times, and I ought to see an (approximately) even distribution of the six faces. Further, if that is not what I observe, I may start to infer some bias in the dice, with varying degrees of statistical confidence.

However when I view the quantum example, would I be right in thinking that it is not possible to directly observe the negative probability of some of the outcomes? That is, I presume if all three paths were open in the quantum case, it would not be possible to observe the ball going down path C –66% of the time.

Actually, the more I think about it with my ‘classical’ head on, the less convinced I am of my presumption. I suppose we must be able to observe something, else how could we conclude that the probability of path C was not –50%, for example? And if we were to add further paths, which result in more negative probabilities within the scenario, how do we detect the contribution of the negative paths? Or am I still failing to comprehend?

I guess some of the above might be explained by the answer to my next question, which is about the total probability of 0% in the quantum case. Is it the case that this probability could have been anywhere between 0% and 100%, and you just happened to choose 0% to help illustrate the weirdness of the maths?

Incidentally, I’m still finding it really hard not to infer some undetected level of interaction in between the red laser and the atom in the Zeno experiment. But I guess I need to keep reminding myself that events at the quantum level are inherently different to what we are used to observing.

Finally, I wonder what the quantum version of the Monty Hall problem would be? And it doesn’t bear thinking what the ensuing debate would be like.:)


This message is a reply to:
 Message 33 by Son Goku, posted 01-04-2007 2:25 PM Son Goku has not yet responded

  
cavediver
Member (Idle past 1806 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 35 of 41 (374898)
01-06-2007 9:36 AM
Reply to: Message 33 by Son Goku
01-04-2007 2:25 PM


Re: Measurement, paths and Zeno freezing.
Hi SG, two points regarding your post that may sound a bit negative but they're really there to spark discussion on how we appraoch this whole area... let me know what you think.

1) I think you've introduced a bit of confusion with your negative probabilities by (unintentionally) suggesting that a specific path can be associated with a -ve prob. I bring forth dogrelate as my first evidence ;)

I must admit I've never liked the "classicalisation" explanations of quantum phenomena (look what happens to this *particle* or consider this *path*) as they perpetuate much of the voodoo nonsense. Quantum behaviour is fascinating but it quickly leads down the road to the confusion we often see expressed by certain individuals here at EvC.

The truth is that classical behaviour is the weird stuff, and quantum the mundane. The fact that a classical regime emerges is what amazes me. And trying to explain the quantum world by reference to the classical is like explaining the chemistry of liquid water by reference to oceanic currents. But can we do better?

2) Now I do like your write up of Zeno freezing... very nice. However, you're sort of playing the magician:

So I've managed to stop an atom from decaying by not interacting with it, in other words, by doing nothing to it.

Flourish, take a bow, curtsy from your beautiful assistant, rapturous applause from the thoroughly baffled and now slightly even more ignorant audiance ;) I think us physicists are so guilty of turning physics into magic just to persuade the masses that it is exciting. I do it all the time...

Think of how you used to view black holes as these truly mystical objects, baffling beyond belief in terms of their ability to screw around with space and time. And then you saw the simplicity of Schwarzschild... were you disappointed or blown away by the elegance and beauty? It is that elegance and beauty I wish we could express better... but could we make anyone else appreciate it?


This message is a reply to:
 Message 33 by Son Goku, posted 01-04-2007 2:25 PM Son Goku has responded

Replies to this message:
 Message 36 by Son Goku, posted 01-06-2007 11:05 AM cavediver has responded

  
Son Goku
Member
Posts: 1121
From: Ireland
Joined: 07-16-2005


Message 36 of 41 (374900)
01-06-2007 11:05 AM
Reply to: Message 35 by cavediver
01-06-2007 9:36 AM


Re: Measurement, paths and Zeno freezing.
Unfortunately I agree with everything you've written, I just have trouble getting this stuff into English. Particularly Zeno freezing. You are also correct to pull me up on it.
I also apologize in advance to dogrelate, I'm going to take cavediver's advice on board and write two much longer posts when I'm satisfied with my attempt to reword my explanations more in the spirit of the actual mathematical material*, so if you think you're confused now, just wait.

*I won't write actual mathematics, just some pseudo-maths to appeal to intuition. I more mean I'll try to cut down on the gee-whiz stuff that leaked into my previous post.

cavediver writes:

1) I think you've introduced a bit of confusion with your negative probabilities by (unintentionally) suggesting that a specific path can be associated with a -ve prob. I bring forth dogrelate as my first evidence


I just don't know how to go about explaining the "real thing". (i.e. Integrals over the space of classical paths). I suppose I was just trying to get across that unlike classical probability more options doesn't necessarily increase the probability. Maybe I should try it from a "action weighted" path point of view?
To be honest, I think this is the mistake I can correct the easiest, since I'm sort of half-way there and simply need to explain that it is the infinite limit of summing over all paths that defines the probability and then explain what is being summed.

Think of how you used to view black holes as these truly mystical objects, baffling beyond belief in terms of their ability to screw around with space and time. And then you saw the simplicity of Schwarzschild... were you disappointed or blown away by the elegance and beauty? It is that elegance and beauty I wish we could express better... but could we make anyone else appreciate it?

First of all, the Zeno freezing is an aspect of (I'm only saying this to set the stage since I know you know it) measurement acting as a projector down onto eigenkets.
The unfortunate thing is how do I explain this physically? I can't use environmental decoherence to make it seem less mysterious, as I would like to, because I can't, considering the ontology of environmental decoherence isn't complete (Problems with the density matrix, e.t.c.). I can't honestly use any of the other interpretations, because they all have their flaws.

So all I'm left with is what happens directly in experiment without any aid from any interpretation or what happens in Dirac's bra ket formalism. I can't use the latter, because that would amount to bringing somebody up to speed on linear algebra and its applicability to spaces of functions.
So all I'm left with is cold, flavourless "turn on the machine and see what happens" experimental fact. Experimentally, light passing straight through an object is doing nothing to it, it is no interaction. And so experimentally the object freezes because of no interaction.
However this is a lame explanation, that can be made less lame by picking an ontology, but which one? And why one above any of the others?
Or am I left only with Dirac?
Or, even worse, I'm I left with only my original explanation, knowing that there is something incorrect about it, but unable to justify a more detailed explanation?

cavediver writes:

I must admit I've never liked the "classicalisation" explanations of quantum phenomena (look what happens to this *particle* or consider this *path*) as they perpetuate much of the voodoo nonsense. Quantum behaviour is fascinating but it quickly leads down the road to the confusion we often see expressed by certain individuals here at EvC.

The truth is that classical behaviour is the weird stuff, and quantum the mundane. The fact that a classical regime emerges is what amazes me. And trying to explain the quantum world by reference to the classical is like explaining the chemistry of liquid water by reference to oceanic currents. But can we do better?


This I have no idea how to do. I know I should do it, but what way would I go about it? All I can see is something like this happening:
Me: So our world isn't really anything but a diagonalization.......jargon, jargon........and also.....jargon, jargon, jargon.......lim (jargon-> infinity)
Person: Eh, what?

(I'm reminded of the time somebody asked my where did electromagnetism come from, what caused it?
How do you get local U(1) symmetry across to somebody and electroweak symmetry breaking?)

Any advice on quantum->classical would be appreciated.


This message is a reply to:
 Message 35 by cavediver, posted 01-06-2007 9:36 AM cavediver has responded

Replies to this message:
 Message 37 by cavediver, posted 01-06-2007 11:50 AM Son Goku has not yet responded

  
cavediver
Member (Idle past 1806 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 37 of 41 (374903)
01-06-2007 11:50 AM
Reply to: Message 36 by Son Goku
01-06-2007 11:05 AM


Re: Measurement, paths and Zeno freezing.
:laugh: I never said it was easy. And please don't take it too hard. We're all in the same boat with this.

However this is a lame explanation, that can be made less lame by picking an ontology, but which one? And why one above any of the others?

Reminds me of picking a gauge... whichever one seem most appropriate to your desired outcome!

Will get back later...


This message is a reply to:
 Message 36 by Son Goku, posted 01-06-2007 11:05 AM Son Goku has not yet responded

  
JavaMan
Member (Idle past 482 days)
Posts: 475
From: York, England
Joined: 08-05-2005


Message 38 of 41 (375280)
01-08-2007 8:55 AM
Reply to: Message 33 by Son Goku
01-04-2007 2:25 PM


Re: Measurement, paths and Zeno freezing.
Thanks for your explanation. I did have some further questions, but having read cavediver's response I realise they're rather pointless :).

However, I would like to help in your attempt to explain the reality of Zeno freezing. So here's my five cents worth.

I guess most of us think we understand the first part of your description:

I will ... consider the case of an atom on its own which starts in a state of having a 50/50 chance of decaying or not decaying.

I'll write the quantum state of the atom in the following form:
(a,b)

Where:
a=current chance of being mesured as decayed.
b=current chance of being measured as not decayed.

So when we start off the atom is in the state (50,50).
I make a measurement to see if the atom decayed. Let's say I find the atom is not decayed, so the state right after measurement is:
(0,100)
Obviously, because I've measured it to be not decayed it has 100% of being not decayed when I measure it.

This doesn't look any different from the case of a tossed coin, having a 50/50 chance of being heads or tails before you've thrown it, and 100% probability of being one or the other once you've tossed it and taken a look. But your next sentence seems to have no analogy in a classical setting:

Then I leave my equipment and stop measuring the atom. Over time the probability returns to 50/50.
So over a time of about a tenth of a second, the following happens to the state of the particle:
(0,100) -> (10,90) -> (20,80) -> (30,70) -> (40,60) -> (50,50).

It sounds to the uninitiated as though the probabilities describe some property or properties of the atom, rather than being statistical effects as they would be in the case of a tossed coin. So my first question is:

Is the state (10,90) a real state of the atom? This question probably appears naive to you and cavediver, but I think understanding what you mean by the state (10,90) is kind of essential for us non-physicists.

P.S. What do you mean in your later post by 'Zeno freezing is an aspect of ... measurement acting as a projector down onto eigenkets'. That sounds like a beginning of an explanation. (Although I'm sure making me understand what an eigenket is will probably try your patience :)).


'I can't even fit all my wife's clothes into a suitcase for travelling. So you want me to believe we're going to put all of the planets and stars and everything into a sandwich bag?' - q3psycho on the Big Bang
This message is a reply to:
 Message 33 by Son Goku, posted 01-04-2007 2:25 PM Son Goku has responded

Replies to this message:
 Message 39 by Son Goku, posted 01-08-2007 3:04 PM JavaMan has responded

  
Son Goku
Member
Posts: 1121
From: Ireland
Joined: 07-16-2005


Message 39 of 41 (375405)
01-08-2007 3:04 PM
Reply to: Message 38 by JavaMan
01-08-2007 8:55 AM


Re: Measurement, paths and Zeno freezing.
Is the state (10,90) a real state of the atom?

Yes it is. You'll see what I mean in a moment.

P.S. What do you mean in your later post by 'Zeno freezing is an aspect of ... measurement acting as a projector down onto eigenkets'. That sounds like a beginning of an explanation. (Although I'm sure making me understand what an eigenket is will probably try your patience ).

This really relates to the question above.
Basically, eigenkets are the "most classical" states of a quantum object. (There is a stricter mathematical definition, but it isn't that important for describing basic QM)
In the example above (0,100) and (100,0) are the eigenkets. As you can see, they are basically "No, I'm not decayed" and "Yes I am decayed".
These are very classical states for something to be in.

Since experimental equipment can only return classical answers (e.g. "Yes I am decayed") these will always be the states after measurement.

Now states like (10,90) are not classical states, they are more generic quantum states that our classical apparatus won't pick up.
So the obvious question following from this would be, what happens if I try to measure the system when it is in a state like this?

As I pointed out above a classical system can only return (0,100) and (100,0), so the system has to jump from (10,90) to (0,100) or (100,0).
(This jump is the projection)
The 10 and the 90 merely label the chance of jumping to each one.

Another way of writing (100,0) and (0,100) would be:
A=(I am Decayed) and B=(I am not Decayed)

In this notation (10,90) would be
(A generic quantum state which will jump to A with a 10% chance and B with a 90% chance).

In this way we label generic states by their chances to jump to the eigenkets following measurement. (10,90) is merely a label for dealing with these generic quantum states, since their chance to jump to the states we can measure is their only important property.


This message is a reply to:
 Message 38 by JavaMan, posted 01-08-2007 8:55 AM JavaMan has responded

Replies to this message:
 Message 40 by JavaMan, posted 01-09-2007 8:15 AM Son Goku has responded

  
JavaMan
Member (Idle past 482 days)
Posts: 475
From: York, England
Joined: 08-05-2005


Message 40 of 41 (375598)
01-09-2007 8:15 AM
Reply to: Message 39 by Son Goku
01-08-2007 3:04 PM


Re: Measurement, paths and Zeno freezing.
Thanks, that was an excellent explanation - I think I understand so far :). Now for some more questions:

1. The Zeno freezing pheneomenon seems to be dependent on there being a relatively slow transition from the measured (0,100) state back to a stable - but not measurable - (50,50) state. Why should there be a slow transition, why doesn't the state just jump back to (50,50) as soon as you stop measuring?

2. What's so special about the classical states? Are they special in the quantum system (does the quantum system somehow know to jump to those states under certain conditions?), or are they only special because the measuring is part of a non-quantum macro world (if that makes any sense :confused:)?

3. Would it be possible under any conditions to directly observe what you call 'generic quantum states'? Will they always be inaccessible to us?

4. My background is in chemistry, so my only knowledge of quantum mechanics is just the very basic stuff you need to understand emission and absorption spectra. Is what you are describing as eigenkets the same phenomenon we observe when we see electrons only ever emitting or absorbing specific quanta of energy? Are the energy levels effectively 'eigenkets' in the terminology you're using? Or is this some different quantum phenomenon?

Edited by JavaMan, : typo


'I can't even fit all my wife's clothes into a suitcase for travelling. So you want me to believe we're going to put all of the planets and stars and everything into a sandwich bag?' - q3psycho on the Big Bang
This message is a reply to:
 Message 39 by Son Goku, posted 01-08-2007 3:04 PM Son Goku has responded

Replies to this message:
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Son Goku
Member
Posts: 1121
From: Ireland
Joined: 07-16-2005


Message 41 of 41 (375817)
01-10-2007 4:14 AM
Reply to: Message 40 by JavaMan
01-09-2007 8:15 AM


Re: Measurement, paths and Zeno freezing.
Your second question is the hardest so I'll leave it to last.

Javaman writes:

1. The Zeno freezing phenomenon seems to be dependent on there being a relatively slow transition from the measured (0,100) state back to a stable - but not measurable - (50,50) state. Why should there be a slow transition, why doesn't the state just jump back to (50,50) as soon as you stop measuring?


It's just the way it evolves. Transitions from states to other states is smooth and slow (relatively speaking).
Basically, similar to how Newton's laws of motion prevent instant movement from one spot to another (you must "slowly" change your position over a period of time), the laws of quantum mechanics prevent instant jumping from state to state (you must "slowly" change your state over a period of time).
The exception to this is the transition to classical states when you make a measurement, when you make a measurement there is a "fast" jump from a generic to classical state.

Javaman writes:

3. Would it be possible under any conditions to directly observe what you call 'generic quantum states'? Will they always be inaccessible to us?


Javaman writes:

4. My background is in chemistry, so my only knowledge of quantum mechanics is just the very basic stuff you need to understand emission and absorption spectra. Is what you are describing as eigenkets the same phenomenon we observe when we see electrons only ever emitting or absorbing specific quanta of energy? Are the energy levels effectively 'eigenkets' in the terminology you're using? Or is this some different quantum phenomenon?


These two questions have related answers, so I'll treat them together.
First of all, yes the energy levels are eigenkets. They are eigenkets of energy, in other words they are classical states where the system can say "Yes I have an energy of x Joules".
(Where x is whatever value you measure.)
Now this brings in the important aspects of QM. The above energy levels are energy eigenkets, states of definite classical energy. There are basically eigenkets corresponding to any quantity you want to measure, for example momentum eigenkets and position eigenkets, e.t.c.

However just because a system is in an eigenket of one quantity doesn't mean it is in an eigenket of another quantity.
An example would be a hydrogen atom. If you measure the energy level of an electron, it now has a definite energy, but since you haven't measured position it is still in a generic state with regard to that quantity.

This is basically where the Heisenberg's uncertainty principle comes in.
The Heisenberg uncertainty principle basically states that you can't be in a position eigenket and a momentum eigenket at the same time.
Or more accurately a eigenket of one is a generic state of the other.

If I have system in a generic state and I measure position, the system jumps to a position eigenket. If I then measure momentum, the system jumps to a momentum eigenket, but measuring momentum forces position to go back into a generic state.
(Or to put it another way a momentum eigenket is generic with respect to position.)
I can never have both of them in classical states at the same time. Measuring one ruins/destroys the "classicality" (to make up a word) of the other.
Hence I can never say where a particle is and what momentum it has at the same time. One quantity will be in a generic state when the other is in a eigenket.

(Sorry to state the same thing over and over, but different people latch on to the concept in different ways, so I'm expressing it as many ways as I can).

Javaman writes:

2. What's so special about the classical states? Are they special in the quantum system (does the quantum system somehow know to jump to those states under certain conditions?), or are they only special because the measuring is part of a non-quantum macro world (if that makes any sense )?


It does make sense and is the opinion of a few physicists. This might be called the "Big" question in Quantum Mechanics, the one that the debates are about. I'll deal with it a separate post shortly.

If anything is unclear, just ask.


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