What is the Meaning of Abstraction in Modern Physics?

[Ken Ucarp]

BELOW IS SOMETHING I COPIED FROM AN OLD THREAD (2004). Would anyone be willing to spend a little time explaining in laymen's terms a few things. In particular, when I as a complete ignoramus read this it sounds like utter mathematical abstraction being used in a manner that seems to posit truth about an actual physical system. Is this pure abstraction or do these things correspond to reality in some way.

START
Suppose that we have two particles, 1 and 2, with corresponding Hilbert spaces H1 and H2. Suppose that particle 1 is in the state |ψ> Є H1, and that particle 2 is in the state |φ> Є H2, and all of this is before the two particles interact. Then, prior to the interaction, the state of the joint system is simply

|ψ>|φ>.

Now, suppose that the interaction between these two particles is such that

|ψ>|φ> → Σk ak|ψk>|φk> ,

where each ak ≠ 0, and there are at least two distinct values for k (and, of course, the |ψk> (|φk>) are linearly independent).

Then, the state of the joint system after the interaction can no longer be written as a simple (tensor) product of one element from H1 with one element from H2 – it must be written as a linear combination of such products. The two particles are now said to be in an "entangled" state.
END

 

[Nugatory]

Is this pure abstraction or do these things correspond to reality in some way?

Here's a question back at you. When you see ##F=ma## do you ask whether that is "pure abstraction" or whether "these things" (##F##, ##m##, and ##a##) correspond to reality in some way?

And having asked that (rhetorical) question, I suggest that the difference between the two cases is in the amount of intellectual effort you have to expend to connect the abstract math to reality. Every high-school student can learn enough math and elementary classical mechanics to connect the ##F=ma## abstraction to reality, whereas it takes a few years of college-level study to do the same with the Hilbert-space abstraction of quantum mechanics. Your reward for expending that effort is that you get to understand more difficult and interesting problems and have a deeper understanding of how the world works.

 

[bhobba]

Here's a question back at you. When you see ##F=ma## do you ask whether that is "pure abstraction" or

I suggest you (meaning the OP - of course Nugatory already has 'gotten' this) study a book on mathematical modeling eg:
https://www.amazon.com/dp/088385712X/?tag=pfamazon01-20

Physics is a mathematical model. What that means regarding the kind of question you asked is not really what this forum or science in general worries about - its more philosophy.

As an example have a look at Newtons laws (Nugatory mentioned the second). Law one follows from law 2 (which is really a definition not a law in the usual sense - it really is a law because its saying forces are important which is a statement about nature - but it isn't an empirical statement type law) and law 3 is equivalent to conservation of momentum which follows from Noethers Theorem and is simply the spatial symmetry of an inertial frame - which is the underlying assumption of Newton's Laws - an inertial frame is assumed. So what is its content? Its actually QM (have a look at the assumptions of Noethers theorem and what I say later to see it) The other explanation is Newtons Laws are a paradigm that says - get thee to the forces. What are forces - simply the concept has been found to help solve classical mechanics problems used by physicists, engineers etc. That's all. In fact as you get more advanced you find yourself using rather than forces, the principle of least action (PLA). Feynman, while still at MIT, when he did advanced mechanics, rebelled against it and applied his exceptional physical talent to solving the problems set using forces, that were really meant to be done using the PLA - he didn't fail - but it made his job much harder than it would have been - probably Feynman was the only one with the ability to do it. The real twist was, via his path integral approach to QM the PLA follows immediately (and that is the answer to what I said before about Noether) - thus explaining why in advanced problems forces often take a back seat - you need a more fundamental law.

So what's the upshot here - mathematical modeling is simply a process of choosing the tools best suited to the task - what they 'mean' sometimes is anyone's guess - at least until a Feynman comes along and explains some of them by deeper models (eg QM), but then you are faced with - what does the deeper model mean? And so it goes.

BTW if you want intuitive understanding of QM at the I level I suggest:
https://arxiv.org/pdf/quant-ph/0101012.pdf

That helps understand the formalism - what it means however is an entirely different matter. If you think that 'bad', 'strange' etc then rest assured people argue about good old probability the same way. The Kolmogorov axioms are about as simple and intuitive as you can get - everyone that studies probability gets it - but what they mean - well that augment, like what QM means is equally as ongoing.

In fact, John Beaz thinks the meaning of QM and probability are related:
http://math.ucr.edu/home/baez/bayes.html
'It turns out that a lot of arguments about the interpretation of quantum theory are at least partially arguments about the meaning of the probability!'

Thanks
Bill

 

[Demystifier]

“Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.”

― Bertrand Russell

 

[Ken Ucarp]

Wow excellent posts, and I have some homework. I'm looking forward to it. And I may not have stated it properly in my OP but I'm not saying, no it can't be this way, obviously I'm not qualified to judge. I'm just saying it seems odd, I understand it's me, and point me in the right direction. Thanks so much. I think too this whole discussion is more along the lines of philosophy of science I guess. Perhaps PF isn't the right place to get into lengthy discussions of this nature. (Not because of the members, but because of the focus).

 

[Ken Ucarp]

Here's a question back at you. When you see ##F=ma## do you ask whether that is "pure abstraction" or whether "these things" (##F##, ##m##, and ##a##) correspond to reality in some way?

And having asked that (rhetorical) question, I suggest that the difference between the two cases is in the amount of intellectual effort you have to expend to connect the abstract math to reality. Every high-school student can learn enough math and elementary classical mechanics to connect the ##F=ma## abstraction to reality, whereas it takes a few years of college-level study to do the same with the Hilbert-space abstraction of quantum mechanics. Your reward for expending that effort is that you get to understand more difficult and interesting problems and have a deeper understanding of how the world works.

This is a great example. F,m, and a all are very close to reality. And even without a mathematical understanding you can sort of "feel" what it means. What you're saying is this same feeling occurs in the more complex stuff but of course only if you have the underpinnings. Makes sense.

I was wondering, if you wouldn't mind just taking that one little piece and delving into it a little more. Namely the opening: particle 1 has a corresponding Hilbert Space. What does that tell me? I think knowing sort of at the base laymens level will help me as I pursue the other links suggested by the other poster.

 

[Nugatory]

This is a great example. F,m, and a all are very close to reality.

It might be more accurate to say that they correspond closely to what your common sense, based on a lifetime of experience, leads you to expect about how macroscopic objects behave when pushed around. That experience is why "even without a mathematical understanding you can sort of 'feel' what it means."

This doesn't work everywhere, and that tells us more about the limitations of common sense than about the relationship between abstract math and the real universe around us.

 

[Leo1233783]

it must be written as a linear combination of such products. The two particles are now said to be in an "entangled" state.
END

it is an experimental fact in physics. However, it is not sure that it is the quantum entanglement of the particles à la EPR experiment.
Here, there is a correlation between the 2 "correlations".

 

[Ken Ucarp]

it is an experimental fact in physics. However, it is not sure that it is the quantum entanglement of the particles à la EPR experiment.
Here, there is a correlation between the 2 "correlations".

But see that's what confuses me. Here's the whole statement: "Then, the state of the joint system after the interaction can no longer be written as a simple (tensor) product of one element from H1 with one element from H2 – it must be written as a linear combination of such products. The two particles are now said to be in an "entangled" state."
So there's a distinction between 'simple tensor product of elements of H' and 'linear combination of such products'. And supposedly these two mathematical ideas represent the reality of a physical situation involving real particles. Now how is it that those same mathematical ideas, or toolsets, (roughly) can also be used to describe say something about biology, or economics? That doesn't make sense. If anything it sounds like somebody is just force-fitting the real world into imaginary but somewhat useful mathematical tools.

 

[Leo1233783]

Ken, I think roughly that the entangled objects are the products and not the particles. I think also that if there is no interaction, it doesn't involve physics but only classical probabilities.

 

[Nugatory]

Now how is it that those same mathematical ideas, or toolsets, (roughly) can also be used to describe say something about biology, or economics?

As far as I know (which isn't very far), they aren't. Hilbert spaces are useful in constructing a theory that properly describes the behavior of subatomic particles, so we use them in that theory. Other problems call for other mathematical tools, and then we use those instead.

If anything it sounds like somebody is just force-fitting the real world into imaginary but somewhat useful mathematical tools.

It goes the other way. We're choosing the mathematical abstraction to fit with what we observe and not the other way around. In this particular example:
All "tensor products of H1 and H2" are also "linear combinations of such products", but not all "linear combinations of such products" are "tensor products of H1 and H2" (people who know what a tensor product is will find that this oversimplification sets their teeth on edge - this can't be helped in a B-level thread). It turns out that those "linear combinations of such products" that are not also "tensor products of H1 and H2" are the ones that describe particles that have interacted and become entangled. But we aren't forcing the world to conform to those linear combinations - instead the world is forcing us to use all the linear combinations instead of just the subset that are also part of the tensor product.

 

[Ken Ucarp]

Interesting. In another thread someone mentioned Hilbert Spaces just being an extension, complex, of matrices, that could be used in other areas of science, not just physics. As far as force-fitting - that makes sense. So what I'm hearing is, the observations are there first. And then the math is used to formalize it.

So is it the case that no matter how obscure and complex the math gets, for example tensor products, Hilbert space this, Hamiltonian that, at some point it has to boil down to numbers being plugged into equations that correspond to observed reality.

 

[Nugatory]

Interesting. In another thread someone mentioned Hilbert Spaces just being an extension, complex, of matrices, that could be used in other areas of science, not just physics

Matrices are indeed a very widely applicable tool. And Hilbert spaces can be thought of (if you squint real hard, skip a lot of steps, and take a casual attitude towards infinite dimensionality) as an extension of matrices, but if so it's an extension that is applicable to a much smaller set of problems.

Compare with the real and complex numbers... The real numbers are such a generally applicable tool that I doubt that there is any problem anywhere in any of the sciences that doesn't use them. The complex numbers are an extension of the reals, but many fewer problems require that we use them.

 

[f95toli]

This is a great example. F,m, and a all are very close to reality. And even without a mathematical understanding you can sort of "feel" what it means. What you're saying is this same feeling occurs in the more complex stuff but of course only if you have the underpinnings. Makes sense.

Note, however, that before the advent of "modern physics" (SR, QM etc) people did indeed worry about what F=ma really "meant" and the discussions were not very different from some of the arguments you see in e.g. threads on PhysicsForums.
The most famous work on this is perhaps Ernst Mach's The Science of Mechanics (which I unsuccessfully tried to read some 20 years ago).
See
https://en.wikiquote.org/wiki/The_Science_of_Mechanics

 

[Ken Ucarp]

Note, however, that before the advent of "modern physics" (SR, QM etc) people did indeed worry about what F=ma really "meant" and the discussions were not very different from some of the arguments you see in e.g. threads on PhysicsForums.
The most famous work on this is perhaps Ernst Mach's The Science of Mechanics (which I unsuccessfully tried to read some 20 years ago).
See
https://en.wikiquote.org/wiki/The_Science_of_Mechanics

Thanks for this reference. I got me started along a line of reading that lead to a paper about Measurement. As I'm reading it seems like that's the key to everything. Philosophers of Science (many obviously who are also scientists) really discuss the very basics. What does it mean to take a measurement and what does it mean to represent reality by mathematical "objects". Fascinating stuff.