## Unity in Multiplicity

Discussions on the nature of being, existence, reality and knowledge. What is? How do we know?

### Re: Unity in Multiplicity

If we look at the nature of "unity" and "mulitiplicity" as phenomenon in themselves, using the example of a point, it can be argued:

1) One point moving towards two+ points causes a seperation that acts simultaneously as division and multiplication. The point however maintains itself as the same point.

****If using an example of a circle or line instead, considering the seperation of a 0d point would equate to 0/x or 0*x unless viewed as an intradimensional structure of 1, the structure forms itself as its own ratio as its own standard of measurement.

2) This movement observes three points if viewed from a structure of 1 dimension. In these respects intradimensional movement starts with 1 as 3.

3) If viewed as progressive and extradimensional, all movement begins quantitatively as 2 and qualitatively under a dualism.

4) This individuation of the one into the many observes a mirroring function where the many mirror the individual nature of the 1 from which they proceed as "units". In these respects multiplicity and "unit" are approximates of "1" and "unity".

5) This approximate nature of the points observes a connection in the respect that the movement from one point to another is dependent on a seperation through direction. In simpler terms, the points seperated because of the direction they moved, hence this direction acts as a linear barrier that:

a) separates by creating a multiplicity as relation.
b) connects as the "movement" itself forms a linear barrier as "direction". Form and function are inseperable, and movement can be observed as the line being conducive to an act of relation.
c) Seperation as projection away from the origin takes a role as a 1d line, while this connection as "movement" takes a role as a -1 dimensional line as a image from which unity is approximated under movement.

6) Division and mulitplication alternate as the manifestation of boundaries with this alternation taking a trinitarian role as "division", "multiplication" and the "alternation" itself.
Eodnhoj7
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### Re: Unity in Multiplicity

mitchellmckain » May 23rd, 2018, 10:12 am wrote:I think I understand what you mean, but the first and last sentences need work in order to be stated correctly. Scientists consider a superposition to be a particular objective state. Say rather that QM tells us that the universe does not consist of point particles with particular positions and velocities. So the billiard ball conception of the universe is certainly unworkable.

It is true that scientists consider a superposition to be a particular objective state. I can agree with a full heart. However, the further problem is that we puny humans have no access to that. Whenever we measure a system, it only presents one of the eigenstates of that superposition to us.

There is the issue of the "eigenstate selection"/"measurement problem"/"wave function collapse" thing. I dunno, for brevity call it ESMPWFC or maybe WFC.

Yourself, or others, could begin to linguistically associate the Universal Wave Function (which contains all the superpositions nicely) as The One Objective Reality. Nothing in physics forbids this reshuffling of terms. In that instance the act of measurement/WFC would be considered a transient phenomenon, a derivative of what is "actually real". Sort of like dipping a drinking straw into an ocean.

While this might seem all very coherent and tidy at first glance, the implications of doing this are certainly not. The room you are in contains transparent glass and objects with certain stable colors. The above framework would be like denying that color exists. To back up a little bit on that briefly -- the reason that anything in the room has color is because electrons can only emit a particular wavelength of light, and the reason is exactly because we only see electrons in the eigenstates of their wave function. This would be like denying that any surface emits color as an objective fact. Whether you like this or not, it is rather forcibly attached to this methodology. It is a corner one paints one's self into, if you have committed to the One Objective Reality is the Universal Wave Function itself.

So in any case --- I hear and understand "Scientists consider a superposition to be an objective state" but they do so in a narrow academic context that is quite divorced from our experience of the universe. I'm not saying this in some emotional way. It is rather concretely divorced. They might have "mathematical reasons" inside the discipline for stating this in narrow academic contexts.

I am not disagreeing with the statement on factual grounds. However, I ask you to consider the narrow context and narrow setting in which that position is adopted.

Furthermore you can grant that the universe is rather machine-like on a large scale, but yes QT does indeed throw a wrench into works of the this conception if you try to take it too far. In particular the rug has been pulled out from beneath Laplace's demon
.
.
.
ar old conceptualization of "material"? It seems to me, our conception of "material" is something that has be constantly changing and with greater rapidity as we approach the present.

I don't know what to make of this bizarre situation that we find ourselves in. On one hand , we begin to try to reach absolute monism in our metaphysics, we are forced, in some way to a singular Universal Wave Function.

But that UWF has pieces we cannot measure. Are they real? David Deutsch says if we "take it literally" then they must be as real as we are. We are forced into the corner with Everettian Many Worlds. Two photons going through two slits in two realities is okay. We can sleep on that. No metaphysical pain is felt. But we are built out of particles. As there are copies of photons - there are copies of us as well

In that corner there are copies of you in other universes. But wait -- that's a plurality! In our desperate attempt to obtain perfect monism, we encounter a kind of extreme plurality. The most extreme : the Many Worlds Interpretation. This situation is like a genie that fulfills wishes, but only at a price. The bigger the wish, the steeper the price. "Genie of the lamp, grant me perfect monism." "It is done." "Wait -- now there are a trillion copies of me in trillion other realities! That's not what I asked for!"

But it was what you asked for. King Midas.

hyksos
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### Re: Unity in Multiplicity

hyksos » May 23rd, 2018, 6:51 pm wrote:
mitchellmckain » May 23rd, 2018, 10:12 am wrote:I think I understand what you mean, but the first and last sentences need work in order to be stated correctly. Scientists consider a superposition to be a particular objective state. Say rather that QM tells us that the universe does not consist of point particles with particular positions and velocities. So the billiard ball conception of the universe is certainly unworkable.

It is true that scientists consider a superposition to be a particular objective state. I can agree with a full heart. However, the further problem is that we puny humans have no access to that. Whenever we measure a system, it only presents one of the eigenstates of that superposition to us.

There is the issue of the "eigenstate selection"/"measurement problem"/"wave function collapse" thing. I dunno, for brevity call it ESMPWFC or maybe WFC.

Yourself, or others, could begin to linguistically associate the Universal Wave Function (which contains all the superpositions nicely) as The One Objective Reality. Nothing in physics forbids this reshuffling of terms. In that instance the act of measurement/WFC would be considered a transient phenomenon, a derivative of what is "actually real". Sort of like dipping a drinking straw into an ocean.

While this might seem all very coherent and tidy at first glance, the implications of doing this are certainly not. The room you are in contains transparent glass and objects with certain stable colors. The above framework would be like denying that color exists.

Not at all. Perhaps what you are missing is the fact that although it is true we only get the eigenstates of the measurement we are making, these are not fixed but entirely relative to the measurement we make. If we measure position, we get a position eigenstate which is a superposition of momentum eigenstates. If we measure momentum, we get a momentum eigenstate which is a superposition of position eigenstates. So your claim that we only see eigenstates is wrong because the eigenstates themselves are superpositions of other eigenstates.

To suggest that some are real and others are not is kind of like saying that only certain times on the clock are real and that when the time reads between them then these are not objective states of time. That would seem rather silly to most people, wouldn't it? But we see the exact same sort of circular continuity in what we call orthogonal measurements. So the problem is a little bit more fundamental than perception by scientists, because this is not a subjective thing at all. If these superpositions are not particular objective states then there would be no particular objective states at all -- and the evidence does not support that either.

hyksos » May 23rd, 2018, 6:51 pm wrote: To back up a little bit on that briefly -- the reason that anything in the room has color is because electrons can only emit a particular wavelength of light, and the reason is exactly because we only see electrons in the eigenstates of their wave function. This would be like denying that any surface emits color as an objective fact. Whether you like this or not, it is rather forcibly attached to this methodology. It is a corner one paints one's self into, if you have committed to the One Objective Reality is the Universal Wave Function itself.

If we talk about science then we are not addressing philosophical questions of reality but the results of measurement. The only commitment involved is following through on what you choose to talk about. If you want to talk philosophy then do so, but this has nothing whatsoever to do with the results of measurements in science.

Ok, perhaps what you means to say is that these superpositions in science suggest to you a philosophical view of reality as lacking in particular objective states. That is not a philosophical viewpoint which I am going to be at all hostile to. Remember, it is my frequent argument that we have no objective evidence that reality is exclusively objective and good pragmatic reasons for believing there is an irreducibly subjective aspect to reality as well. Isn't this is practically saying the same thing in different words? If so, your previous hostility to these words is a little confusing. However, what I will be hostile to is the effort to claim that science proves such a philosophical outlook, because it does not (just because I believe it doesn't mean science proves it). When it comes to the subjective and the philosophical, a diversity of thought is an unavoidable reality we must accept.

hyksos » May 23rd, 2018, 6:51 pm wrote:So in any case --- I hear and understand "Scientists consider a superposition to be an objective state" but they do so in a narrow academic context that is quite divorced from our experience of the universe.

Not hardly! Since they do so in a context which is directly connected to experiments which can demonstrate the fact, your claim this is divorced from our experience of the universe is a bit bizarre!

hyksos » May 23rd, 2018, 6:51 pm wrote:They might have "mathematical reasons" inside the discipline for stating this in narrow academic contexts.

The mathematics simply predicts the results of physical measurements.

hyksos » May 23rd, 2018, 6:51 pm wrote:I am not disagreeing with the statement on factual grounds. However, I ask you to consider the narrow context and narrow setting in which that position is adopted.

If you are wondering what this has to do with life as experienced in some other context (whether it is drinking in a bar or the daily tasks of a social worker), then I am inclined to ask how the objects such as electrons and photons have their role in these activities. While science is focusing on individual electrons and photons, everyday life only experiences them averaged en masse.

hyksos » May 23rd, 2018, 6:51 pm wrote:
Furthermore you can grant that the universe is rather machine-like on a large scale, but yes QT does indeed throw a wrench into works of the this conception if you try to take it too far. In particular the rug has been pulled out from beneath Laplace's demon
...
ar old conceptualization of "material"? It seems to me, our conception of "material" is something that has be constantly changing and with greater rapidity as we approach the present.

I don't know what to make of this bizarre situation that we find ourselves in. On one hand , we begin to try to reach absolute monism in our metaphysics, we are forced, in some way to a singular Universal Wave Function.

What I learn from science is the value of looking at things from different perspectives so why in the world would I insist on wearing my mathematical glasses and insist on seeing everything through those lenses all the time? Thus I simply find this way of looking at things useful but do not feel compelled by it as you seem to be.

hyksos » May 23rd, 2018, 6:51 pm wrote:But that UWF has pieces we cannot measure. Are they real? David Deutsch says if we "take it literally" then they must be as real as we are. We are forced into the corner with Everettian Many Worlds. Two photons going through two slits in two realities is okay. We can sleep on that.

Because these "other worlds" are not measurable, then this is not a matter of objective scientific results but of subjective interpretation. Another interpretation, as I have explained in the other thread, is that these are nothing but real possibilities which did not actually happen.

hyksos » May 23rd, 2018, 6:51 pm wrote:In that corner there are copies of you in other universes. But wait -- that's a plurality! In our desperate attempt to obtain perfect monism, we encounter a kind of extreme plurality. The most extreme : the Many Worlds Interpretation. This situation is like a genie that fulfills wishes, but only at a price. The bigger the wish, the steeper the price. "Genie of the lamp, grant me perfect monism." "It is done." "Wait -- now there are a trillion copies of me in trillion other realities! That's not what I asked for!"

I have bit different take on this. The Everett interpretation is the result of making the mathematics perfectly deterministic and the result is this plurality which treats all these possibilities as real things which actually happen. But think about it. Is this not a logical consequence of presupposing determinism in a world which is not deterministic?
Last edited by mitchellmckain on May 23rd, 2018, 10:56 pm, edited 1 time in total.

mitchellmckain
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### Re: Unity in Multiplicity

So your claim that we only see eigenstates is wrong because the eigenstates themselves are superpositions of other eigenstates.

Our communication has severely broken down. An eigenstate is most assuredly not a superposition, at all. It is practically speaking the opposite of one. Perhaps we are reading different textbooks here. Would you prefer that I ditch "eigenstate" and instead use the phrase "stationary state"?

Because these "other worlds" are not measurable, then this is not a matter of objective scientific results but of subjective interpretation. Another interpretation, as I have explained in the other thread, is that these are nothing but real possibilities which did not actually happen.

What I interpreted you were saying was something like this : "Look at all these esteemed scientists who say that a superposition is an objective state. Now shouldn't you agree be in agreement with all these esteemed scientists?"

Now your words have completely flip-flopped to the other side. If your former belief was "if we can't measure it then its a matter of subjective interpretation" then you should have said this from the beginning, rather than beating me over the head with a laundry lists of all the esteemed scientists who say "superposition is objective."

I am perfectly happy to describe "what is measured" as being objectively real. But let it be known, we only ever measure particular sharp eigenstates of the wave function anytime anything is measured. By our collectively-agreed-upon definition, these eigenstates must be what is objective. Frankly I don' tknow what this conversation is pulling me into, but by no means do I intend to say that that if the continuously-evolving wave function (happening between measurements) is not objective, ergo it must be subjective.

That is not true at all whatsoever. I suppose what I need to to to shore up any kind of sanity in this conversation is to stop using the word "objective" (because that attracts mitchmckain to me too strongly) and instead replace it with some phrase like "Stable physical Elements of reality".

perhaps what you means to say is that these superpositions in science suggest to you a philosophical view of reality as lacking in particular objective states. That is not a philosophical viewpoint which I am going to be at all hostile to.

Yes. I'm merely only attacking a very narrow mechanistic view here. But I will warn you this is not personal between me and you. There is rather, an army of Hidden Variable Theorists out there. You are one person, but the sweaty HVT advocates are huge in number. Experimental physics in the last 18 years has been bothered enough by them to actually perform experiments to debunk them. Which means, despite whatever you are proselytizing on this forum, people are NOT spending millions of dollars to debunk you. They are spending millions of dollars to debunk HVT with very expensive lab equipment and random number generators built in Austria, and atomic clocks, and et cetera. These experimetns are being carried out in Colorado, in Japan, likely parts of Europe , and likely other continents.

"Very Good Reasons" I see you have adopted bandying about this phrase "very good reasons".

Well sciencechatforum, when mitchel mckain uses that phrase, in 90% of the cases in which he does use that phrase his "very good reasons" are in fact only some other posts he made "in another thread" on this forum.

In contrast, when I say "very good reasons" I'm talking about million-dollar experiments performed by teams of scientists in Japan, Colorado, Italy, and Austria. With statistical validity, and fully published results in esteemed peer-reviewed journals.

Remember, it is my frequent argument that we have no objective evidence that reality is exclusively objective and good pragmatic reasons for believing there is an irreducibly subjective aspect to reality as well. Isn't this is practically saying the same thing in different words?

Absolutely not. As far as I'm concerned, this is your own personal theory. I will presume our breakdown in communication is because you see the word "objective" and believe it is taking on a definition that is used in humanities departments. But I am using "objective" in a very narrow sense from physics. I am not in any way referring to any objective/subjective split or rift, or whatnot.

When I say "objective" I mean to communicate something much closer to : stable physical elements of reality.

(Another way of saying this - -) If you encoded the physical state of a system as a bit string, then that bit string would always either have a 1 or a 0 in each position, and there would exist a clean and absolute 1-to-1 correspondence between the bit encoding and the physical state.

If I could get us to rise above the clouds ...

It should be repeated and emphasized in neon letters : There do exists very prescient philosophical debates raging in actual physics departments, quite disconnected from this forum and the people on it, and their local, provincial squabbles. Let me go back to my original post and quote myself from where I entered this thread, if I may.

hyksos » May 22nd, 2018, 12:10 pm wrote:The metaphysical grist of Quantum Theory does not pivot on a inescapable plurality. Rather the metaphysical pivot point in QM is that it suggests that the universe is not in a particular objective state at time t. The universe is not a grand machine made of "gears" whose orientations are objectively determined at any given instant. Even if the universe were "one singular substance" (lets say) that substance would not be in a particular configuration/state at time t.
I have no reason to back down from nor amend this paragraph. This debate is very real. Even if not a single poster on this forum is engaging it, it still exists, and research scientists are still throwing big money at it.

This debate goes all the way back to the EPR paper where Albert Einstein himself argued that quantum mechanics must be incomplete. He then wrote several famous letters to Max Born, where he first said (1) "..at any rate I do not believe that God throws dice." A few decades later, he wrote to Born again, saying (2) "You and your students believe in the God which throws dice, but I do not."

If the million-dollar experiments to test Bell's Inequalities in Austria, Colorado, Japan, and Italy are not good enough for you -- then add to that -- the greatest physicists known to western civilization debated this aspect of the world.

hyksos
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### Re: Unity in Multiplicity

(Just to do some closet-cleaning here)

Being an HVT does not make you stupid, nor uneducated. I do describe them as "Sweaty" and "puffing cigarettes" on occasion -- but only as a way of communicating my building frustration with their obstinance. I don't know where the regular forum posters fit in this debate.. or if this forum is chock-full of HVTs who don't quite know it yet, and maybe haven't given the issue much thought.

At the heart of the HVT is this sentiment : In order for there to be a universe at all, it must be in a particular state at time t, the alternative is senseless, anti-scientific, and fails the Principle of Sufficient Reason.

Albert Einstein had this sentiment. Podolsky and Rosen had it. Ironically, John Bell held this sentiment. (Yes, the guy we named the Bell's Inequalities after). John Bell himself was an HVT advocate !!

On example of this would be mitchelmckain who says something pretty close to "There exists an irreducible subjective aspect of the world." ( While this may not be the time or place to debate the veracity of that sentence ) let us agree temporarily that it is true. Say the universe contains an irreducible subjective element. There is still a strong sense that the Subjective Element, however it may actually be, would still be in a particular state S at a time t. If that subjective element changed state, you could still, in principle encode that state as a bit string, and the bit string would be 'coherent' in the sense of always ever a 1 or 0 in each position. Always and ever a 1-to-1 correspondence between the encoding the state S , and of the Subjective Element instantiated in physical space.

Quantum Mechanics is suggesting this is all completely wrong. It says the universe we inhabit is not like this. It says that if you were to encode any physical system, those bits would maybe be a one or a zero in some positions. But what are they "really"? They are really neither state, extended in space and time neither and both. Not just the encoding, but the actual extant physical particle is in both positions. This cuts deep.

Albert Einstein was bothered by this. How bothered? He was so bothered he knuckled down with two colleagues and published a fricking paper on it -- a paper demanding that reality and the universe cannot be like this. Like he didn't just meet him for a cup of tea to "discuss it". He got out a typewriter and got busy. Now you and I can post stuff on the internet. Sure. but when a scientists of international esteem like Einstein was, writes a paper on something, that is the scientific equivalent of declaring war.

Like people get divorced over this stuff. This is the metaphysical equivalent of Protestants and Catholics in Northern Ireland. Friendships end and furniture is broken over this issue.

Roughly speaking what has been going on since about 2004 are experiments to test Bell's Inequalities. The crux of this is that any suggested physical mechanism proposed by an HVT cannot describe the physics of how entangled particles cause each other to lose superposition.

Another crux woudl be the God-throwing-dice issue. The HVT cannot describe what electrons are doing in ultracold superconductors, when their energy is zero. They not only continue to move, but continue to change configurations in a way that alters their total entropy. This is explicitly forbidden by classical thermodynamics. Things at zero energy cannot "reconfigure". (Hint : they do, and they have. In labs.)

hyksos
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### Re: Unity in Multiplicity

I am going to focus on just this part until we get it clear.

hyksos » May 23rd, 2018, 9:50 pm wrote:
So your claim that we only see eigenstates is wrong because the eigenstates themselves are superpositions of other eigenstates.

Our communication has severely broken down. An eigenstate is most assuredly not a superposition, at all. It is practically speaking the opposite of one. Perhaps we are reading different textbooks here. Would you prefer that I ditch "eigenstate" and instead use the phrase "stationary state"?

Your understanding is incomplete. I can explain it but how can I prove it to you? I would quote my text by Sakurai if that would help. The online pdf is here and you want section 1.4 pages 23 to 36. But it is a graduate textbook and so perhaps isn't helpful to you. Here is something from a Proffessor Mark Alford.
7. The uncertainty principle. Now you can see (qualitatively) how the
uncertainty principle arises. The eigenstates of one operator are not in
general the same as the eigenstates of a different operator. So when you
make a state with a definite value of one observable, it will in general not
have a definite value of the other observables. For example, a plane wave,
which is an eigenstate of the momentum operator, has a definite value of
the momentum. But plane waves are completely spread out in space, so the
plane wave contains all eigenstates of the position operator, so the position
is then completely uncertain

The eigenstates are relative to the measurement operator. Pick a different measurement operator and the eigenstates are different even in the same measurement space. Thus the eigenstates of one operators will be combinations (superpositions) of the eignstates of the incompatible (non-commuting) operator.

Let's try a different example. How about fermionic particle spin. with this you have to measure the spin in a particular direction. Measure it in the x direction and the eigenstates are either +x or -x. Meaure it in the y direction and the eigenstates are either +y or -y. So what happens if you first measure in the x direction and then in the y direction? Your first measurement depends on what state you start in. But after the first measurement we know the spin is either +x or -x and both of these give a probability of 50% +y and 50% -y. So you see... the eigenstate with respect to one measurement is a superposition with respect to a different measurement.

Like the example of position and momentum, these are an example of measurement operators which are non-commuting and this ties in with the Heisenberg uncertainty principle which is a consequence of this. Because the eigenstates of one operator is a linear combinations of the eigenstates of the other operator you cannot measure both at the same time and we call them non-commuting because you get different result depending on which you measure first.

Also a correction is needed in my previous post
To suggest that some are real and others are not is kind of like saying that only certain times on the clock are real and that when the time reads between them then these are not objective states of time. That would seem rather silly to most people, wouldn't it? But we see the exact same sort of circular continuity in what we call orthogonal measurements. So the problem is a little bit more fundamental than perception by scientists, because this is not a subjective thing at all. If these superpositions are not particular objective states then there would be no particular objective states at all -- and the evidence does not support that either.

"orthogonal" is the wrong word. It should be "non-commutative."

mitchellmckain
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### Re: Unity in Multiplicity

The eigenstates are relative to the measurement operator. Pick a different measurement operator and the eigenstates are different even in the same measurement space. Thus the eigenstates of one operators will be combinations (superpositions) of the eignstates of the incompatible (non-commuting) operator.

This is totally wrong. I will show you exactly where your error is.

"Thus the eigenstates of one operators will be combinations (superpositions) of the eignstates of the incompatible (non-commuting) operator."

What does "will be" indicate here? Are you defining the word eigenstate to be equal to superposition? That is most certainly not correct.

I will agree to this syllogism :

If> .. a given observable `A` has "collapsed" to an eigenket by an operator `Y`,

Then> ..`A` will appear concomitantly in physical reality, along with a superposition of the observable `B` which is non-commuting with `A`.

That syllogism was written by me, from my understanding. You will notice it is in exact agreement with all the textbooks you have quoted, included Sakurai's book.

You have not made that claim given by this syllogism. Instead what you have claimed is that an eigenstate IS a superposition. (is? You said "will be". What are you saying?) That is not-so-clearly wrong and there cannot exist any matrix where this would be the case (excluding trivial examples like zero matrices).

I guess my main complaint here is you keep saying "Eigenstates are superpositions of eigenstates." That's pretty much what this boils down to. That's a peculiar bastardization of english language. It barely makes any semantic sense. That would be like saying "A tree is a collection of trees."

I guess would I further say that to refer to this squirrely phrase
combinations (superpositions) of the eignstates

Why would anyone care about such an animal like that? This phrase only serves to confuse readers, confound forum moderators, and is so bizarre as to indicate a profound misunderstanding of the words by the writer. Nothing in particular should stop a person from referring to "combinations of eigenstates" -- I am not claiming such a strange beast makes no sense.. I certainly understand what you mean by this phrase, but it is very unclear what the intent here is. For example, you could measure a system over and over again, and write down all the measurements on a piece of paper, then later in the day, point to the paper and refer to a "combination of eigenstates". Sure. But what is the point?

Because the eigenstates of one operator is a linear combinations of the eigenstates of the other operator you cannot measure both at the same time and we call them non-commuting because you get different result depending on which you measure first.

This is even worse. Now instead of One eigenstate is one superposition, you are referring to eigenstates (plural) is a linear combination of eigenstates (plural). This is bizarre and going nowhere!

Bottom line: An (single) eigenstate is not a (single) superposition.

Period.

hyksos
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### Re: Unity in Multiplicity

mitchellmckain,

You have succeeded in chasing this entire thread down into minutiae involving the meanings of individual words in textbooks.

Did you have anything at all to say about any of the actual content of my posts?

Do you have anything to say about Hidden Variable Theories?

Do you have anything to further the conversation regarding the experiments performed on HVT?

Do you believe that HVT and entanglement are irrelevant to the metaphysics of holism? Can you defend that position?

I guess I'm asking if you have any meat and potatoes to bring to this table, or if this is just going to turn this thread into acting like proof-readers for quantum mechanics textbooks. Your call.

hyksos
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### Re: Unity in Multiplicity

What is the point in discussing any science with you when you simply refuse to accept the facts of that science. I was open to working out the problems (going to textbooks on your suggestion hoping that would get you to listen), but apparently you are not willing. Since you insist on simply dictating confused rhetoric contrary to the science then, no, I have no interest in this any more than I have any interest in the flat earth society or creationism.

The only thing I have to say about hidden variable theories, is that they have been experimentally proven wrong.

Entanglement is certainly relevant to holism in metaphysics. In fact, even special relativity has some implications in that direction - an interesting combination of both separation and contectedness.

I don't serve meat and potatoes. I cook a diverse cuisine of Thai, Mexican, Indian, Italian, Chinese, Russian, Korean, and British. Perhaps my philosophical offerings are equally diverse and too much for the sensitive digestion of those more accustomed to simple fare.

Bottom line: An (single) eigenstate is not a (single) superposition.

Bottom line as stated by Prof. Mark Alford: While the eigenstates of a single measurement operator is not a superposition with respect to that measurement operator, they often are a superposition of the eigenstates of a different incompatible (non-commuting) measurement operator. The example given by both myself and Prof Alford is that the eigenstates of the momentum operator are a linear combination of all the eigenstates of the position operator, and visa versa. And the point was that you making some kind of absolute difference between eigenstates and superposition states is just plain WRONG!

mitchellmckain
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### Re: Unity in Multiplicity

The differentiation between eigenstate and superposition is quite rigid, and it has to do with matrices. There is actually something in a matrix called an eigenvalue. They represent the Schroedinger wave as a "state" in a Hilbert Space and give a matrix for that state, wherein each entry in the matrix is a complex number.

For a given matrix, there will be an eigenvector, and that eigenvector is called an "eigenket" in quantum mechanics textbooks. This sentence may be entirely true :

the eigenstates of the momentum operator are a linear combination of all the eigenstates of the position operator, and visa versa.

That sentence in isolation may be completely and totally true. Even mathematically so. It's interesting too. (It is interesting enough to dig up the equation for it!)

But it it is far far too specific to glean a DEFINITION of "eigentstate". The word eigenstate in is a very broad term in quantum mechanics, and while momentum is certainly an example of a particular eigenstate observable, it is a by no means the only example. There is also an eigenstate for angular momentum, which if observable, will correspond to what is regularly called a "spin state". There are eigenvalues of a wave matrix that are not even observable at all.

I cannot emphasize enough how broad "eigenstate" is. It is broadest way possible to refer to an aspect of Matrix Mechanics that comes up in this universe. If I wanted to avoid the word, and risk "Talking down" to an audience, I would say something else like "standing wave". While "standing wave" would make a lay audience nod in understanding, a physics geek in the room would be cringing because the word choice does not adequately capture the nuances of QM.

Nevertheless, a standing wave is the simplest, clearest pedagogical example of an "eigenstate" money can buy. When you enhance and upgrade your education to textbooks, these simple examples must be dispensed with and replaced by more all-encompassing mathematical procedures. You upgrade from 2D waves to vectors in infinite dimension Hilbert Spaces. You must then depart from simple PBS Nova TV explanations and start dealing with things like eigenvalues of a matrix. These more abstract , more foundational tools will capture a much broader range of physical phenomena.

During unitary wave evolution (when the system is not observed, measured, or probed) that the vector in the Hilbert space representing the Schroedinger wave is definitely NOT an eigenvector. Consequently you could never claim that the state the wave is in at that time is an eigenstate. It is vehemently not an eigenstate, and furthermore, it could not be a "linear combination" of them either.

hyksos
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### Re: Unity in Multiplicity

I almost forgot.
mitchellmckain » May 27th, 2018, 9:17 am wrote:The only thing I have to say about hidden variable theories, is that they have been experimentally proven wrong.

Amen.

Perhaps my philosophical offerings are equally diverse and too much for the sensitive digestion of those more accustomed to simple fare.

That's fine. For myself, I tend to always eat from the salad bar.

hyksos
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### Re: Unity in Multiplicity

hyksos » May 27th, 2018, 4:36 am wrote:The differentiation between eigenstate and superposition is quite rigid, and it has to do with matrices. There is actually something in a matrix called an eigenvalue. They represent the Schroedinger wave as a "state" in a Hilbert Space and give a matrix for that state, wherein each entry in the matrix is a complex number.

For a given matrix, there will be an eigenvector, and that eigenvector is called an "eigenket" in quantum mechanics textbooks.

Yes the operators can be matrices which operate on a space of vectors. The eigenvectors are those in this space which give a particular eigenvalue in the eigenvalue equation. But the point here that you keep glossing over is that the all of this is completely relative to which matrix you have in your eigenvalue equation. With a different matrix (a different measurement operator in QM) you get a different set of eigenvectors and different eignvalues to go with them. And while the eigenvectors of the one matrix are independent (not a linear combination of the other vectors), they are a linear combination of the eigenvectors of the OTHER matrix.

so matrix A
2 1 0
1 2 1
0 1 2
gives eigenvalues with eigenvectors:
2 with a1 = (1 0 -1),
2+sqrt(2) with a2 = (1 sqrt(2) 1)
2-sqrt(2) with a3 = (1 -sqrt(2) 1)
BUT a different matrix B
1 2 0
3 2 3
1 0 1
gives you these eigenvalues with eigenvectors:
1 with b1 = (0 0 1)
-1 with b2 = (1 -1 -1/2)
4 with b3 = (2 3 2/3)
You can normalize these eigenvectors if you want but it is not required for this demonstration.
So for example we can see that b1 = .25 a2 + .25 a3 - .5 a1
Thus we see that the eignevectors of one matrix operator is a linear combination of the eigenvectors of a different matrix operator.

hyksos » May 27th, 2018, 4:36 am wrote:But it it is far far too specific to glean a DEFINITION of "eigentstate". The word eigenstate in is a very broad term in quantum mechanics, and while momentum is certainly an example of a particular eigenstate observable, it is a by no means the only example. There is also an eigenstate for angular momentum, which if observable, will correspond to what is regularly called a "spin state". There are eigenvalues of a wave matrix that are not even observable at all.

I cannot emphasize enough how broad "eigenstate" is. It is broadest way possible to refer to an aspect of Matrix Mechanics that comes up in this universe. If I wanted to avoid the word, and risk "Talking down" to an audience, I would say something else like "standing wave". While "standing wave" would make a lay audience nod in understanding, a physics geek in the room would be cringing because the word choice does not adequately capture the nuances of QM.

NO! The terms in science and mathematics are VERY specific -- there is nothing "broad" about them. Ok, sure. You appropriate the terminology of science for a pseudo-scientific gobble-dee-gook for a lot of subjective philosophy in such things as motivational speaking or selling products in some pyramid scheme. But I hope you don't seriously expect everyone to buy into such nonsense!

hyksos » May 27th, 2018, 4:36 am wrote:Nevertheless, a standing wave is the simplest, clearest pedagogical example of an "eigenstate" money can buy. When you enhance and upgrade your education to textbooks, these simple examples must be dispensed with and replaced by more all-encompassing mathematical procedures. You upgrade from 2D waves to vectors in infinite dimension Hilbert Spaces. You must then depart from simple PBS Nova TV explanations and start dealing with things like eigenvalues of a matrix. These more abstract , more foundational tools will capture a much broader range of physical phenomena.

During unitary wave evolution (when the system is not observed, measured, or probed) that the vector in the Hilbert space representing the Schroedinger wave is definitely NOT an eigenvector. Consequently you could never claim that the state the wave is in at that time is an eigenstate. It is vehemently not an eigenstate, and furthermore, it could not be a "linear combination" of them either.

Switching to a different set of scientific and mathematical term which you want to appropriate for your rhetoric is only going to get this reply: "fool me once shame on you, fool me twice shame on me." I am hardly going to waste the time digging into another set of math and science terms see if there is any substance there, when the first such effort led nowhere.

mitchellmckain
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### Re: Unity in Multiplicity

mitchellmckain » May 27th, 2018, 2:53 pm wrote:BUT a different matrix B
1 2 0
3 2 3
1 0 1

The matrix B from which I derived those eigenvalues and eigenvectors is....
1 2 0
3 2 0
1 0 1

While I am typing... how about the commutator AB - BA?
5 6 0 . 4 5 2
8 6 1 - 8 7 2
5 2 2 . 2 2 2
which as you can see is not zero so these are non-commuting operators, which hardly a surprise for matrix operators.

Anyway if the matrices commute then they basically have the same eigenvectors, though the eigenvalues can be different. If one of the matrices has a degeneracy, two or more of its eigenvectors are usually chosen somewhat arbitrarily. In that case, you can just generalize the claim to say that the eigenvectors CAN be chosen the same when it commutes with another matrix. For example, the identity matrix commutes with all other matrices, and all of its vectors are chosen arbitrarily. But then they can be chosen as the same as the eigenvectors of any other matrix.

mitchellmckain
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### Re: Unity in Multiplicity

mitchellmckain » May 27th, 2018, 11:53 pm wrote:Yes the operators can be matrices which operate on a space of vectors. The eigenvectors are those in this space which give a particular eigenvalue in the eigenvalue equation. But the point here that you keep glossing over is that the all of this is completely relative to which matrix you have in your eigenvalue equation. With a different matrix (a different measurement operator in QM) you get a different set of eigenvectors and different eignvalues to go with them

This is absolutely true mathematically. But in no shape or form did I "gloss over" this.

And while the eigenvectors of the one matrix are independent (not a linear combination of the other vectors), they are a linear combination of the eigenvectors of the OTHER matrix.

This is true in some instances -- but in no shape or form is this the definition of "eigenvector". This is why I was so careful to highlight in pink your incessant use of "is" and "Will be".

so matrix A
2 1 0
1 2 1
0 1 2
gives eigenvalues with eigenvectors:
2 with a1 = (1 0 -1),
2+sqrt(2) with a2 = (1 sqrt(2) 1)
2-sqrt(2) with a3 = (1 -sqrt(2) 1)
BUT a different matrix B
1 2 0
3 2 3
1 0 1
gives you these eigenvalues with eigenvectors:
1 with b1 = (0 0 1)
-1 with b2 = (1 -1 -1/2)
4 with b3 = (2 3 2/3)
You can normalize these eigenvectors if you want but it is not required for this demonstration.
So for example we can see that b1 = .25 a2 + .25 a3 - .5 a1
Thus we see that the eignevectors of one matrix operator is a linear combination of the eigenvectors of a different matrix operator.

These are real-valued elementary matrices. Okay good. Now you can get an A- on a 12th grade proficiency test taken by high school kids. Great.

Let it be known -- the matrices in quantum mechanics have complex numbers in all of those locations (instead of real numbers) and some of them have infinite dimensions. While elementrary real-valued matrices are entirely required, you will need to perform quite a bit of abstraction to reach the Matrix Mechanics of QM.

NO! The terms in science and mathematics are VERY specific -- there is nothing "broad" about them. Ok, sure. You appropriate the terminology of science for a pseudo-scientific gobble-dee-gook for a lot of subjective philosophy in such things as motivational speaking or selling products in some pyramid scheme. But I hope you don't seriously expect everyone to buy into such nonsense!

Absolutely nothing I have typed is gobble-dee-gook, psudo-scoence or product selling.

The word "eigenstate" in QM is broad in the sense that it is the GENERALIZATION of the concept of "standing wave" abstracted to higher dimensional waves expressed as operators on complex number spaces. In other words , "standing wave" is an elementary , provincial example of an "eigenstate" for a wave traveling on a 2D string.

Eigenstate is broad in another sense -- It captures the mathematics of a much larger set of physical phenomenon (including for example "spin states" which cannot exist on physical strings, at all).

If you doubt the veracity of what I have said here, invite Lincoln to examine this post. I am confident that he will confirm the veracity of everything I have posted.

Switching to a different set of scientific and mathematical term which you want to appropriate for your rhetoric is only going to get this reply: "fool me once shame on you, fool me twice shame on me." I am hardly going to waste the time digging into another set of math and science terms see if there is any substance there, when the first such effort led nowhere.

Not what is happening here, Mr. Mckain.

Take a wave on an elementary string and express its state as a matrix. Find the eigenvectors of that matrix. Any of those eigenvectors will correspond with the modes of a standing wave on that string. I.e. the speed the wave is propagating is a whole number of the length of that string. You will get stationary nodes. (Hence the phrase, stationary state ). https://en.wikipedia.org/wiki/Stationary_state

Repeat this calculation on a complex-valued matrix. Choose a matrix that represents rotations (call it an "operator" if you want to sound academic.) You will be repeating the exact calculation that is used in QM textbooks that predicts the existence of quantized angular momentum.

You want page numbers?

hyksos
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### Re: Unity in Multiplicity

hyksos » May 27th, 2018, 11:44 pm wrote:You want page numbers?

Nah... Bored now. I think I am just going to say "whatever," and move on.

Oh and there is no implication here that you are boring. It is more about me being bored with the role I am playing in this discussion. So... you can say that I am boring, if you like.

mitchellmckain
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### Re: Unity in Multiplicity

There was no other direction to go, other than to start working individual problems out of the back of the chapters of a quantum mechanics textbook.

That someone (anyone) succeeded in drawing a thread that far into a digression -- that should raise yellow flags for both moderator and forum user alike.

hyksos
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