Non-Commutation Property and its Relation to the Real World

[expos4ever]

TL;DR Summary

It seems mysterious to me that properties of mathematical structures appear to dictate the nature of reality

I just learned that if two linear operators do not commute, this means when we use operators to characterize observables in quantum mechanics, the corresponding observables cannot both be definite at the same time. This seems hard to believe to me since I have a strong intuition, perhaps mistaken of course, that "mathematical structures" should not have the power to "dictate" the nature of reality. Let me be more concrete. I am told that the operators for momentum and position do not commute and that this means that, if I understand the argument properly, that nature itself will not allow a "particle" to possess both a position and a momentum simultaneously. I do not understand why these operators do not commute and, perhaps if I did, the mystery would disappear. But, at the moment, I believe, repeat believe, that I am being told that the properties of linear algebra determine something very fundamental about nature. And that seems wildly counterintuitive.

By the way, I am not sure if this post belongs in the linear algebra forum (here) or in the quantum mechanics forum. Moderators will move it, I guess, as they see fit. I also get the impression that this forum does not want "philosophical" matters to be discussed. However, I hope the following paragraph makes it clear that I am trying to stay grounded in the domain of physics.

In any event, am I right to be so astonished? Is there a sense in which the properties of mathematical objects (in this case commuting linear operators) actually dictate the way the world is? Or do I have the cart before the horse? Perhaps one of you will tell me that the reason the linear operators for position and momentum do not commute is because they are "set up that way by we humans" because we already know, from experiments, that these two observables cannot simultaneously have definite values. In that case, the astonishment is far less - there would be no sense in which mathematical structure dictates reality. However, even if experimental evidence determines the nature of these linear operators, it still seems remarkable that linear algebra can so closely model reality. Perhaps, though, this strays into the possibly forbidden territory of philosophy.

 

[strangerep]

Perhaps one of you will tell me that the reason the linear operators for position and momentum do not commute is because they are "set up that way by we humans" because we already know, from experiments, that these two observables cannot simultaneously have definite values.

This. To understand it a little more deeply, consider: how would you create a particle in a state such that it had both definite momentum and definite position (to perfect accuracy)?

In that case, the astonishment is far less - there would be no sense in which mathematical structure dictates reality. However, even if experimental evidence determines the nature of these linear operators, it still seems remarkable that linear algebra can so closely model reality. Perhaps, though, this strays into the possibly forbidden territory of philosophy.

A famous quantum physicist Eugene Wigner was also puzzled about this, and wrote an essay titled "The Unreasonable Effectiveness of Mathematics". You can easily find it via google.

The bottom line is that mathematics is just logic. We choose particular mathematical tools in various circumstances that account closest for known phenomena. The best theories then predict new, hitherto unknown, phenomena, showing that we have a reaonably faithful correspondence between the mathematical model and physical reality.

 

[Dale]

TL;DR Summary: It seems mysterious to me that properties of mathematical structures appear to dictate the nature of reality

I have a strong intuition, perhaps mistaken of course, that "mathematical structures" should not have the power to "dictate" the nature of reality.

Do you also believe that the JSWT created all those stars and galaxies? If not then why would you make your absurd statement that I quoted? It is a really absurd statement. Are you just “trolling”, saying something you know is bizarrely unreasonable just to get some traction?

 

[malawi_glenn]

"mathematical structures" should not have the power to "dictate" the nature of reality

Let me use a quote from Bohr, but slightly modified: "Stop telling nature what to do".

Why do you not have these objectives against Newtonian dynamics?

 

[martinbn]

I do not understand why these operators do not commute and, perhaps if I did, the mystery would disappear.

If you know what operators are, what these operators in particular are, and what it means to commute or not, then this should be clear. If not, then you should first make sure you understand the definitoins of those terms.

But, at the moment, I believe, repeat believe, that I am being told that the properties of linear algebra determine something very fundamental about nature. And that seems wildly counterintuitive.

I think it have it backwards. It is the the properties of nature that tells us what the mathematical objects that model it should be. But if you have a good model, then the consequences of that model should also describe something about reality. For example, Euclidean geometry is a good model for small scale gemetry on Earth. Then a pure maths theorem (Pythagoras') says that if you take two sticks of length 3 and 4 units, put two ends at a right angle, then the distance between the other two will be 5 units. And if you do the experiment it is so. Do you find this counterintuitive?

 

[PeterDonis]

I believe, repeat believe, that I am being told that the properties of linear algebra determine something very fundamental about nature.

No, you're not. You're being told that something about nature--something that has been tested in many experiments--is described in math using linear algebra with non-commuting operators. The math isn't dictating reality; reality is dictating the math.

 

[PeterDonis]

I do not understand why these operators do not commute

Because that's how nature works at the quantum level: many actual measurements do not commute--their results depend on the order in which you make them. So to properly describe nature in this domain, we need to use math that has that property.

 

[expos4ever]

Thanks to those who answered courteously and with respect. However, I have decided to leave this forum so I will not be reading any more replies. Thanks again.

 

[hutchphd]

TL;DR Summary: It seems mysterious to me that properties of mathematical structures appear to dictate the nature of reality

Or do I have the cart before the horse?

Does it matter i.e. does $$\left [ horse,cart \right]=0~?$$

 

[malawi_glenn]

If you know what operators are, what these operators in particular are, and what it means to commute or not, then this should be clear. If not, then you should first make sure you understand the definitoins of those terms.

I think it have it backwards. It is the the properties of nature that tells us what the mathematical objects that model it should be. But if you have a good model, then the consequences of that model should also describe something about reality. For example, Euclidean geometry is a good model for small scale gemetry on Earth. Then a pure maths theorem (Pythagoras') says that if you take two sticks of length 3 and 4 units, put two ends at a right angle, then the distance between the other two will be 5 units. And if you do the experiment it is so. Do you find this counterintuitive?

Or that the angular sum in a triangle is 180 degrees. Well that does not work on large scales of the surface of the Earth, we need to change math then!

 

[hutchphd]

Or that the operations of rotation and translation do not commute?? Compare rotation by 90deg and walking a mile to walking a mile and rotating 90deg. The difference is √2 mile!

 

[weirdoguy]

However, I have decided to leave this forum

Well, as much as I like PF the way it is, I think this happens way to often.

 

[malawi_glenn]

Well, as much as I like PF the way it is, I think this happens way to often.

What do you think could have been made different in this case? From what I see, OP was given good replies. Not only in this thread, but in all threads of theirs.

OP was last seen 4 h after their latest post. I think OP will return :)

 

[expos4ever]

What do you think could have been made different in this case? From what I see, OP was given good replies. Not only in this thread, but in all threads of theirs.

OP was last seen 4 h after their latest post. I think OP will return :)

What do you think could have been different? See post 3.

I have been back on this site primarily to exchange some PMs. If you guys want this to be a place where people who are not as knowledgeable as most of you are feel welcome, well, I think you could be a tad more charitable. Let me be clear: it is not just the one post that I am concerned with. It is the fact that, based on things I have learned outside this thread, there appears to be a belief that such rudeness is "the price of doing business", as it were.

I do not buy that for a second.

In any event, I may indeed stick around, but only because my interest in physics eclipses my dignity.

 

[hutchphd]

In my experience most physicists value the truth above all else. Personally I consider it a mark of great respect if someone goes to the often considerable trouble of figuring out why I am wrong. I treat such comments as gifts, and. where I come from "you don't look a gift horse in the mouth".
This is particularly the norm in technical matters and may seem rude to the uninitiated. It truly is not intended that way, but if you ask a truly ill-considered question, don't expect a soft rebuke and your dignity is not really the point. But good open minded inquiries are always welcomed.

 

[expos4ever]

It truly is not intended that way, but if you ask a truly ill-considered question, don't expect a soft rebuke and your dignity is not really the point. But good open minded inquiries are always welcomed.

This is a kinder, gentler form of exactly the thing I am talking about. You, of course, have zero evidence that my question was ill-considered. It may seem that way from your perspective, but it is patently unfair to imply anything of the sort on my part.

My question was genuine, I got flamed (if that is the right word) and you are making it seem this was deserved. And yet you clearly have no grounds for such a conclusion.

A politely worded question deserves an answer, not a rebuke, soft or otherwise. And to say "my dignity is not the point" is further evidence of the problem here - I would very much consider the dignity of the other person in any reply I would offer.

 

[expos4ever]

In my experience most physicists value the truth above all else. Personally I consider it a mark of great respect if someone goes to the often considerable trouble of figuring out why I am wrong. I treat such comments as gifts, and. where I come from "you don't look a gift horse in the mouth".
This is particularly the norm in technical matters and may seem rude to the uninitiated. It truly is not intended that way, but if you ask a truly ill-considered question, don't expect a soft rebuke and your dignity is not really the point. But good open minded inquiries are always welcomed.

...and given that so many have applauded this dismissive post, I am done. If your goal was to scare off someone who wants to learn, mission accomplished.

 

[PeterDonis]

You, of course, have zero evidence that my question was ill-considered.

The question itself is evidence. Please note that "ill-considered" and "genuine" can both be true of the same question. You intended the question as an honest question; you say so, and I'm willing to take you at your word. But there was an obvious alternative (the one I described in posts #5 and #6 and that others gave examples) to the "astonishing" inference that you drew, that you did not consider. If another poster thinks you should have considered that obvious alternative, then to them, your question is ill-considered.

Empathy goes both ways. If you are asking for empathy for your honest attempt at asking a question, you should be willing to grant empathy to others' honest descriptions of how they see the question.

 

[strangerep]

[...] I may indeed stick around, but only because my interest in physics eclipses my dignity.

I have been where you are now. Many rapid learning experiences for me have involved my dignity getting trashed. Eventually, I learned to stow my ego in the basement and just concentrate on trying to learn. Becoming proficient in some area of maths+physics is an excellent way to restore one's inner sense of dignity.

HTH.

 

[martinbn]

@expos4ever I am confused. What exactly in the replies caused you such grief?