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Can Fractals Make Sense of the Quantum World?
Keith found a New Scientist story about fractals and quantum theory. The article says "Take the mathematics of fractals into account, says Palmer, and the long-standing puzzles of quantum theory may be much easier to understand. They might even dissolve away."
Well, the point of the article is that if the underlying structure of the universe is fractal, then it shows why, for instance, you can measure the position or the velocity of an electron, but not both; the general idea is that instead of a linear reality, the universe exists along a fractal edge, and answers derived using current quantum methods are literally falling off the edge because they're not finely enough resolved - they don't take the foaminess of the edge into account, so they miss the answer and land in a space that literally isn't part of the real universe - they're undefined. This is an illuminating and interesting idea, and it may point directly to how we could measure both at the same time, which would make a lot more sense to some of us. Me included.
He's not incorporated all of quantum theory into his fractal idea, so this is far from certain, but it is a lovely idea.
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Fractals are basically the incorporation of decisions into iteratively applied functions of some kind. Physics normally uses mathematics of varying degrees of curves and shapes and spaces to describe things and these functions are continuous to a degree, and so its pretty reasonable to think that such descriptions could be imprecise. Math tends to see "switch and loop and jump" statements as inelegant and those are the essence of fractals.
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An old Canadian friend's brother turned out to be a mathematical physicist working at a Canadian university researching fractal spacetime. Garnet Ord's work supposedly reconciles the notoriously conflicting relativity and quantum mechanical models of spacetime. It seems that the time axis used to be treated as an integer variable, when in fact it's a fractional dimension: a fractal.
I'd say that making relativity and QM interoperate is a good way to "make sense of the quantum world".
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So if I understand this correctly, Palmer is saying that the universe has a finite amount of information variables and at some certain point it will reach that limit? And that every time we try a thought experiment to measure either the position or a velocity of a particle, we risk overstepping that finite limit and thus get results where we can only measure one or the other because to do both sets us beyond the limit? So then can it be inferred that he's saying the universe has a limit then?
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The problem is that Bell's Thm. tests for hidden variables - essentially "deeper physics".
And Bell's Thm. has been verified repeatedly.
So, either he's arguing that Bell's Theorem is taking us down a blind alley, or he's going to have to figure out someway to make both the fractal understanding and Bell's true. The article in New Scientist doesn't discuss that at all.
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>The article was pretty vague handwaving. It didnt actually how any problem was solved with fractal mathematics. It could have tried to explain one example.
By coincidence I just looked through a text book on 'quantum chaos' today, paying attention to an example they had for the quantum mechanics of the Helium atom. (something I know something about, as a chemical physicist).
What they did there, was model Helium semi-classically as 'colinear'; as if the two electrons and the nucleus were in a straight line. A pretty weird model from a physics standpoint, but I suppose necessary from their perspective since that dynamical system apparently displays chaotic behavior. After some math, they managed to show how this replicated the overall spectrum of Helium.
Now that's nice and fairly impressive. But I don't actually see any direct usefulness of it. It's not a better or more accurate way than solving the Schrödinger equation for the electrons. It does illustrate that the main properties of atoms/molecules are due to the nonlinear dynamics of electron motion. But we knew that already. So in a way it was like a lot like how you react to fractals: "Well, that does look a lot like a fern leaf!... So?"
Now I'm not entirely certain if this is representative of the work in TFA. But there's a definitely the risk when you attempt to mate 'buzzword topics' like this, that you start doing stuff for its own sake, and always end up with rather contrived connections. Now, if chaos theory can really explain quantum physics at a more fundamental level, that's one thing. But I don't think coming up with chaotic systems that share properties with quantum ones is doing so, any more than a fractal image of a fern leaf 'explains' the biology of ferns.
The article loses me almost immediately when it states that information is lost in a black hole. Anyone who's read Susskind's book knows that this implies all sorts of unpleasantness like the irreversibility of the the S-matrix, and so it is likely incorrect; ie, information is not lost when objects fall into a black hole. This makes sense, because to an outside observer, an object never falls into a black hole, it only approaches the event horizon without ever quite reaching it. Therefore, one would expect that information from objects falling into a black hole is written on the surface of the event horizon. This represents the highest information density possible. This is Susskind's thesis, and it was my understanding that it is becoming the accepted view. Stephen Hawking was a proponent of black-hole information loss, and Palmer was a student of Hawking (20 years ago). Therefore, it is not surprising his theory is based on rejected premises.
This is an illuminating and interesting idea, and it may point directly to how we could measure both at the same time, which would make a lot more sense to some of us. Me included.
I'm good with not being able to directly determine position and velocity simultaneously. The part I have problems with is the position and velocity uncertainty also applies to nothingness. The more sure you are that an area of space contains no particles, the more uncertain you are how fast they are going.
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Well, the point of the article is that if the underlying structure of the universe is fractal, then it shows why, for instance, you can measure the position or the velocity of an electron, but not both ... This is an illuminating and interesting idea, and it may point directly to how we could measure both at the same time, which would make a lot more sense to some of us. Me included.
IMHO thinking of the position, velocity, energy and lifetime of particles is a hard way to go about things in and of itself. Whilst it's a correct interpretation of quantum mechanics, it's also just as correct to think of everything as waves, which I find easier.
Thinking in this way an electron is simply a wave, as is a photon, and so on. Multi-particle systems are just combinations of these waves added together, and Fourier analysis shows that even these individual waves are just a combination of simple sine waves. (In quantum mechanics these simple waves are the allowed 'eigenstates', which lets us forget about 'particleness').
Now take a regular sine wave. What is its amplitude (energy)? It's exactly 1, with no uncertainty. However, such a sine wave is infinitely long. As nothing can go faster than light, the wave would have to spread out for an infinite amount of time to become infinitely long. If we apply the energy/time uncertainty principle to this wave we get:
uncertainty in energy * uncertainty in lifetime > a constant
We have no uncertainty in energy, but the uncertainty in the wave's lifetime is infinite, so by a little non-rigorous argument we can say that 0*infinity could well be more than a constant. That's obviously very dodgy maths, but it's just an analogy since infinite waves don't exist. Now let's make the wave realistic.
To do this we have to make a finite wave, ie. it must start and it must stop. Since any wave can be made from a sum of other waves we just need to add up an infinite number of ever-smaller waves which cancel out the main one before the start and after the end. The result is a sine wave which grows then shrinks, fitting into a finite length. But now what is its amplitude? That question's harder than before, since it depends on what time you look at it, and thus the uncertainty in the time you take. Also, how long does it live for? Well since we're summing an infinite series of waves it never really cancels fully, it just gets smaller and smaller, so the uncertainty in lifetime depends on how well you know the amplitude.
If we put these two properties together we can deduce that there's no way for us to know both at once, since they depend on each other. Since we're only allowed to use certain waves in our sum (the eigenstates mentioned before) the sum is still infinite, but there are steps between the waves. It is therefore straightforward to say:
uncertainty in energy * uncertainty in time > some constant to do with the allowed waves
This is the energy/time uncertainty principle, with the constant being Planck's constant / 4 pi.
A similar line of reasoning can be used for the frequency of a wave in a given length, to obtain the velocity/position uncertainty principle.
I feel you're on to something here: Along the lines of the parent post I put on... let's assume that fractals are correct and that Palmer's right. Would that then mean that there is a limit to the universe, in terms of using fractals to make sure we get the calculation just right to avoid 'hitting nothing' when calculating position and velocity? If so, is non-existence quantifiable? Or does the act of measuring it increase existence? My head is starting to hurt here, so I'd like to ask if someone far more knowledgable than I am can answer this. What I'm thinking though, is that if Palmer's correct, then we might have found an edge of the universe (so to speak), and if we have, then wouldn't that put us a whole lot closer towards determining whether or not we are in a simulation (a better way to put it would be : we are the simulation?)
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