Can the 'shape' of the universe be described at the Plank scale?

[BobCastleman]

By way of introduction, I am a database engineer looking at returning to graduate school to focus on the overlaps in information theory, quantum theory and relativity. So I'm working my way through undergraduate texts on the foundations of said theories, attempting to resurrect long dormant math skills, etc.

Needless to say, I have many questions, and as I'm not at University, few resources for getting answers.

Many of my questions are at this point conceptual not mathematical, so I apologize for my lack of rigor. But this is important to me as I need to build appropriate visualizations before the math makes sense to me.

So two very basic questions:

Does it make sense to talk about the "shape" of the universe at the Plank scale? I mean this in the same sense that at the macro scale the universe is thought to be spherical, flat or hyperbolic.


I understand that the position of an electron in its orbital cloud is essentially probabilistic and you can't talk about where it is at one moment and then predict where it will be in the next moment. It's next position could be anywhere within the cloud based on the probability function. So my question: Does it make sense to describe the motion of the electron as discontinuous? E.G. in classical mechanic, you can't get from point A to point C without passing through point B. But in a probabilistic model, there is a non-zero chance of "appearing" at point C.

Thank you for your patience.

Bob

 

[ZapperZ]

By way of introduction, I am a database engineer looking at returning to graduate school to focus on the overlaps in information theory, quantum theory and relativity. So I'm working my way through undergraduate texts on the foundations of said theories, attempting to resurrect long dormant math skills, etc.

Needless to say, I have many questions, and as I'm not at University, few resources for getting answers.

Many of my questions are at this point conceptual not mathematical, so I apologize for my lack of rigor. But this is important to me as I need to build appropriate visualizations before the math makes sense to me.

So two very basic questions:

Does it make sense to talk about the "shape" of the universe at the Plank scale? I mean this in the same sense that at the macro scale the universe is thought to be spherical, flat or hyperbolic.

What exactly is meant by the "shape of the universe" at that scale? In fact, what does it mean at ANY scale? The "shape" of something means that there's a boundary signifying when the object has an abrupt end. How do you ask that at a microscopic scale that is inside the object itself? Does this makes any sense to you?

I understand that the position of an electron in its orbital cloud is essentially probabilistic and you can't talk about where it is at one moment and then predict where it will be in the next moment. It's next position could be anywhere within the cloud based on the probability function. So my question: Does it make sense to describe the motion of the electron as discontinuous? E.G. in classical mechanic, you can't get from point A to point C without passing through point B. But in a probabilistic model, there is a non-zero chance of "appearing" at point C.

Thank you for your patience.

Bob

You have to be very careful here. Once you have made a measurement, the description of the system is now different than what it was before. The act of measurement has put the system into a particular state, and the original description is no longer valid.

The "motion" of the electron isn't continuous, simply because we cannot track such a movement. If we simply go by the wavefunction, then we can already see that it isn't discontinuous based on such a description.

While it is tempting to do quantum mechanics before understanding the mathematics, relying purely on physical intuition isn't something I highly recommend. What grounds QM to what we already know IS the mathematics. The physical description of it is discontinuous (since you are interested in such a concept) from our classical physical intuition. So to start with that ahead of the mathematics will often result in a misunderstanding of what QM is.

Zz.

 

[BobCastleman]

What exactly is meant by the "shape of the universe" at that scale? In fact, what does it mean at ANY scale? The "shape" of something means that there's a boundary signifying when the object has an abrupt end. How do you ask that at a microscopic scale that is inside the object itself? Does this makes any sense to you?

Perhaps "shape" is the wrong word. Maybe the correct word is geometry? Parallel lines behave differently depending on the geometry of the space in which they reside.

"Does such a geometry exist at the Planck scale?" might be the better question.

You have to be very careful here. Once you have made a measurement, the description of the system is now different than what it was before. The act of measurement has put the system into a particular state, and the original description is no longer valid.

Having made a measurement, can we say with any certainty what the next measurement will be? This "feels" a little like a Markov Process - the measurement "erases" any previous description (a Markov Process requires no knowledge of previous state) and the next state is wholly independent of that description. But the next state is still probabilistic.

The "motion" of the electron isn't continuous, simply because we cannot track such a movement. If we simply go by the wavefunction, then we can already see that it isn't discontinuous based on such a description.

(hopeless naivete ahead) If our ACTUAL measurements are essentially discrete then how can we verify that ANY description of continuous motion is valid? The wave function is a model that suggests continuity but that continuity cannot be verified with actual measurements?

While it is tempting to do quantum mechanics before understanding the mathematics, relying purely on physical intuition isn't something I highly recommend. What grounds QM to what we already know IS the mathematics. The physical description of it is discontinuous (since you are interested in such a concept) from our classical physical intuition. So to start with that ahead of the mathematics will often result in a misunderstanding of what QM is.

Zz.

I am in no way avoiding the mathematics. I'm neck deep in it. I just work on multiple tracks and one of those is not so much developing an intuitive model so much as correcting the intuitive (or abandoning) model as needed.

 

[ZapperZ]

Perhaps "shape" is the wrong word. Maybe the correct word is geometry? Parallel lines behave differently depending on the geometry of the space in which they reside.

"Does such a geometry exist at the Planck scale?" might be the better question.

No one knows.

Having made a measurement, can we say with any certainty what the next measurement will be? This "feels" a little like a Markov Process - the measurement "erases" any previous description (a Markov Process requires no knowledge of previous state) and the next state is wholly independent of that description. But the next state is still probabilistic.

That's the whole point of "wavefunction collapse"! You have put the system in a particular state. If I measure the spin of an electron, and finds that it is in a particular direction, and I continue to isolate it from everything else, what do you think is the outcome of my next, identical measurement of it?

[qutoe](hopeless naivete ahead) If our ACTUAL measurements are essentially discrete then how can we verify that ANY description of continuous motion is valid? The wave function is a model that suggests continuity but that continuity cannot be verified with actual measurements?[/quote]

But just because your measurement is discrete also doesn't make the system discrete! So the same argument can be applied to you.

I have a system that is described by, say, a plane wave. Can you tell me how, from that description, that I have a discrete system for, say, it's position? Again, unless you can argue this from the point of view of the mathematics, you really have nothing to stand on if you simply want to argue this based on "intuition".

I am in no way avoiding the mathematics. I'm neck deep in it. I just work on multiple tracks and one of those is not so much developing an intuitive model so much as correcting the intuitive (or abandoning) model as needed.

Read above. You simply cannot put the cart before the horse. Rather than looking at the actual object itself, you are already making guesses on what it is based on simply looking at the shadow of the object.

Zz.

 

[BobCastleman]

No one knows.

Seems like an obvious question. Is the geometry at Planck scale irrelevant or just not yet determined?


(from your previous response - emphasis mine)

If we simply go by the wavefunction, then we can already see that it isn't discontinuous based on such a description.

But just because your measurement is discrete also doesn't make the system discrete! So the same argument can be applied to you.

If measurement is discrete how does one verify actual continuity? Seems that even if the wave function describes continuous motion, there is no form of measurement that allows us to verify that.

 

[Ken G]

Seems like an obvious question. Is the geometry at Planck scale irrelevant or just not yet determined?

Not yet determined, in the worst way. The best theory we have now (GR) isn't just moot on the question, it gets what must be the wrong answer to the question-- it's not consistent with quantum mechanics. This question awaits unification of QM and GR.

If measurement is discrete how does one verify actual continuity? Seems that even if the wave function describes continuous motion, there is no form of measurement that allows us to verify that.

Yes, I would say an important lesson of quantum mechanics is that we are not allowed to "mentally fill in the gaps" between our measurements, like we thought we could do in classical mechanics (notice what happens, for example, in the double slit experiment if you try to do that). Instead, quantum mechanics introduces a fundamental role for indeterminacy-- instead of what is determined and what is not determined being an issue of little importance, as it was in classical mechanics, in quantum mechanics it is a very important part of the complete picture. One might even say that quantum mechanics involves the union of what is determined from what is indeterminate, and both occupy essential places in the patchwork. Continuous motion was always a kind of fantasy of classical mechanics that we got away with but could never really verify. In quantum mechanics, it is part of the laws that we cannot verify it.

 

[ZapperZ]

Seems like an obvious question. Is the geometry at Planck scale irrelevant or just not yet determined?

Why is this an "obvious" question? Obvious to whom? And what's with this Planck scale stuff when you're not even anywhere close to understanding QM, much less, QFT?

If measurement is discrete how does one verify actual continuity? Seems that even if the wave function describes continuous motion, there is no form of measurement that allows us to verify that.

Think of classical physics. Applying what you just said, even classical physics, using your logic, is "discrete". Is this something you are arguing for?

Zz.

 

[BobCastleman]

Why is this an "obvious" question? Obvious to whom? And what's with this Planck scale stuff when you're not even anywhere close to understanding QM, much less, QFT?

Well it was obvious to me. And apparently not irrelevant based on Ken G's response.

Think of classical physics. Applying what you just said, even classical physics, using your logic, is "discrete". Is this something you are arguing for?

Measurement is discrete? No? At some point any form of ACTUAL measurement has some unit value that beyond which measurement has no meaning.

I'm not arguing for anything. I'm just asking questions. If measurement is the limiting factor in the validation of ANY physical theory, then anything that cannot be measured cannot be verified. Thus continuity cannot be verified. This does not argue that the universe is continuous or discontinuous, only that we cannot determine it to be one way or the other.

 

[tommyli]

Does it make sense to talk about the "shape" of the universe at the Plank scale? I mean this in the same sense that at the macro scale the universe is thought to be spherical, flat or hyperbolic.

Hi Bob, I think this is a very fair question. I've also heard of things like this from TV documentaries on string theory. I think those who have already commented are much more qualified to talk about this since it is something I know very little about, but I think I can offer a simple suggestion for a physical picture that explains what other people have already said, which is that nobody knows.

If you think of usual X-ray crystallography: it is possible to study structure of a crystalline material because the wavelength is comparable to the lattice spacing. Now if instead of X-rays you shine physical light, you clearly see nothing of the internal structure, because the wavelength is much larger than the distance between atoms (the internal structure is 'smeared out'). If you care about long-wavelength behaviour, for example the long-range wavefunction of an electron, the electron appears to move in free space, and the only effect of the crystal is to give the electron a different mass (in condensed matter we call this the effective mass approximation). This is valid when you are dealing with distance scales much larger than the distance between atoms.

Now it turns out that you can do calculations in this approximation by writing the Hamiltonian for a particle that moves in vacuum with some effective mass. Instead of looking at the crystal, all information about the electron's interaction with the atomic cores and other electrons is buried in the new "mass" of the electron (the renormalized mass).

For a single electron, this picture is good, but as soon as you go to field theory, you find that as soon as you calculate anything, you get an infinite answer, because you must integrate over all momenta. Usually your integral diverges as momentum becomes infinite. There is a clear physical interpretation: when momentum becomes large, this corresponds to a distance scale (via the de Broglie relation) which is smaller than the spacing between atoms. So you have to introduce a cutoff (the ultraviolet cutoff), p_max at some "high" value of momentum to prevent your answer from becoming infinite. The interpretation is that in the theory you do not probe at distances less than h-bar/p_max.

Now for a crystal this is very sensible. However it turns out that when you do field theory in a vacuum, you also get similar infinities. "Common sense" would say that you can go to arbitrarily small distance scales and the vacuum should not have any structure there, there should be no problem when momentum becomes very large, so people do not understand the origin of these infinities. So people introduce a bit of mathematical machinery to get around these infinities. You say that there is a cutoff momentum beyond which you do not integrate, and then you always get a finite answer. However if you decide to interpret this physically, you can say that the momentum corresponds to a length scale which is the scale on which the gravitational force dominates, and possibly something disruptive is happening there, and the momenta over which you do integrate do not account for gravity at all.

 

[BobCastleman]

However it turns out that when you do field theory in a vacuum, you also get similar infinities. "Common sense" would say that you can go to arbitrarily small distance scales and the vacuum should not have any structure there, there should be no problem when momentum becomes very large, so people do not understand the origin of these infinities. So people introduce a bit of mathematical machinery to get around these infinities.

Your entire response was helpful, but this part jumped out at me. The limits to what we can probe and the mathematical machinery to deal with the infinities is very interesting.

FWIW, my motivation behind these questions lies in information systems and how they map to physics. Our computer representations are fully discrete, our measurements have "granularity" that is easily modeled as discrete and even our mathematics are discrete strings of symbols. My interest in Planck scale is that it is essentially a lower limit to what can be validly symbolically represented in something like a Turing Machine or a Deterministic Finite Automata.

 

[ZapperZ]

Why is this an "obvious" question? Obvious to whom? And what's with this Planck scale stuff when you're not even anywhere close to understanding QM, much less, QFT?

Well it was obvious to me. And apparently not irrelevant based on Ken G's response.
Measurement is discrete? No? At some point any form of ACTUAL measurement has some unit value that beyond which measurement has no meaning.

No, your measurement, ANY measurement, is "discrete". Show me what you think is a continuous measurement, and I'll show you a discrete one.

I'm not arguing for anything. I'm just asking questions. If measurement is the limiting factor in the validation of ANY physical theory, then anything that cannot be measured cannot be verified. Thus continuity cannot be verified. This does not argue that the universe is continuous or discontinuous, only that we cannot determine it to be one way or the other.

But unless you argue that nature puts something very strange in between our measurements, there is a physical description of a continuous evolution from one to the other.

For example, in a plane wave state, the particle has a continuous probability to be in all those places. Now, while measurements will reveal a particle being at x1, x2, x3, etc... unless you are arguing that there's something strange going on in between x2 and x2, then from the description that we have (the wavefunction), one can say that the position probability is continuous between x1 and x2, and so on, even when our measurements produces values at discrete locations.

Again, this is also what is done in classical systems. It is just that it is so closed together, it APPEARS as if it is a continuous set. It isn't!

And no, we DO know when there's a discontinuity. The existence of phase transition is a clear example. So it is not as if we don't know if there is a breakdown in the continuous evolution of a parameter. We do, and know if one occurs!

Again, show me an example of what you think is continuous. What you claim is not supported by the physics. That is the main problem that I have with what you are claiming.

Zz.

 

[BobCastleman]

Again, show me an example of what you think is continuous. What you claim is not supported by the physics. That is the main problem that I have with what you are claiming.

Zz.

Maybe I've given the wrong impression. I'm not trying to claim that the universe is or isn't continuous.

I'm concerned with computer representations of physics. Continuity CANNOT be represented in a computer. And it seems to go deeper - that continuity is an abstraction that cannot be verified by ANY measurement. So while continuous functions are extraordinarily powerful, both descriptively and predictively, they cannot be linked to ACTUAL observation.

 

[Ken G]

I actually think you are making a very valid point, which is that in classical physics, we had discrete measurements but imagined that they were just tracing out something fundamentally continuous. This never created a problem in classical physics (though encounters subtleties in nonlinear dynamics), but it does in quantum physics-- what is continuous in quantum mechanics is the wave function, but the wave function implies certain inherent indeterminacies. So in a sense, it is what is indeterminate that is continuous, but what is determinate cannot be demonstrated as being continuous. The closer you look for continuity, the more you mess up the result! Verifying continuity is a bit like opening a book to verify that it was closed.

So we can address this in two ways-- rationalistically, we say that if the wave function is evolving continuously, then so is "nature", and it is only our ability to verify this fact that is the problem. But empirically, we say that what we can verify is all we can claim to be true, so if we cannot verify continuity, then we cannot claim it either. That is more along the lines of the way you are thinking, and I think it's a fine angle of approach. We really don't know at this stage which angle will bear the most fruit down the road (such as in investigations of the vacuum like tommyli was insightfully describing), so exploring every avenue is the rational course.

Indeed, it sounds like you are interested in how we manipulate finite amounts of (therefore discrete) information, so even if nature is "really" continuous, it is never going to matter to us that it is-- what will matter to us will always be the finite information we actually have access to. So it makes sense to model information as discrete, so that we are talking about the same kinds of information as what we actually get access to. Underlying that information might be a continuous theory, just as it was in classical mechanics, but that was never what we were actually dealing with-- which has important implications even in classical physics, in the context of chaotic systems.

I think what ZapperZ is saying is that quantum mechanics by itself really doesn't have answers to your questions, it is a mathematical structure that is fully self-contained and answers what it answers (basically, statistical predictions), but leaves open what it leaves open (what vacuum is, whether finite information is an inherent aspect of how nature works or if it is just a limitation we face in exploring nature, etc.). Since you are talking about going beyond quantum mechanics, it might be a good idea to master the mathematics of quantum mechanics first, while perhaps keeping your eye out for potential insights into the questions that really interest you. I think developing a working philosophy, while you learn the math, is a good idea-- it can really help channel your learning into fruitful directions, and increase your pleasure in a process that can otherwise be a bit dry if not daunting.

 

[BobCastleman]

I actually think you are making a very valid point, which is that in classical physics, we had discrete measurements but imagined that they were just tracing out something fundamentally continuous. This never created a problem in classical physics (though encounters subtleties in nonlinear dynamics), but it does in quantum physics-- what is continuous in quantum mechanics is the wave function, but the wave function implies certain inherent indeterminacies. So in a sense, it is what is indeterminate that is continuous, but what is determinate cannot be demonstrated as being continuous. The closer you look for continuity, the more you mess up the result! Verifying continuity is a bit like opening a book to verify that it was closed.

So we can address this in two ways-- rationalistically, we say that if the wave function is evolving continuously, then so is "nature", and it is only our ability to verify this fact that is the problem. But empirically, we say that what we can verify is all we can claim to be true, so if we cannot verify continuity, then we cannot claim it either. That is more along the lines of the way you are thinking, and I think it's a fine angle of approach. We really don't know at this stage which angle will bear the most fruit down the road (such as in investigations of the vacuum like tommyli was insightfully describing), so exploring every avenue is the rational course.

Indeed, it sounds like you are interested in how we manipulate finite amounts of (therefore discrete) information, so even if nature is "really" continuous, it is never going to matter to us that it is-- what will matter to us will always be the finite information we actually have access to. So it makes sense to model information as discrete, so that we are talking about the same kinds of information as what we actually get access to. Underlying that information might be a continuous theory, just as it was in classical mechanics, but that was never what we were actually dealing with-- which has important implications even in classical physics, in the context of chaotic systems.

I think what ZapperZ is saying is that quantum mechanics by itself really doesn't have answers to your questions, it is a mathematical structure that is fully self-contained and answers what it answers (basically, statistical predictions), but leaves open what it leaves open (what vacuum is, whether finite information is an inherent aspect of how nature works or if it is just a limitation we face in exploring nature, etc.). Since you are talking about going beyond quantum mechanics, it might be a good idea to master the mathematics of quantum mechanics first, while perhaps keeping your eye out for potential insights into the questions that really interest you. I think developing a working philosophy, while you learn the math, is a good idea-- it can really help channel your learning into fruitful directions, and increase your pleasure in a process that can otherwise be a bit dry if not daunting.

You captured the gist of my thinking here. I bolded the parts that were particularly salient.

As for mastering the math of quantum mechanics, I fully understand this as a necessity, else everything is so much speculative blather. I just learn on parallel tracks. The math has to grow along side the conceptualization. The math tells me which parts of the conceptualizations need adjusting or must be abandoned, but I need both to progress.

 

[Ken G]

Yes, this is why many of us chose to be physicists, rather than mathematicians-- that "parallel track" that is so important to many physicists.