"The idea of a continuous range, so familiar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is accessible to us."

   I've had similar suspicions myself. At the most fundamental levels, nothing is smooth. There is no connectedness, in other words.

   Do you ever wonder what reality looks like? At best, we can only approximate. In fact, it might not look like anything, since the building blocks of reality are smaller than photons. But, we are judging the ability to see in terms of what we consider visible light.

   Currently reading "The Road to Reality", if you're curious.

I have no idea what you mean by connectedness . . . That all sounds like metaphysical debate, something that I am not quite at the level to comprehend quite yet. Break it down for me!

I could spend 10 minutes or 10 years trying to explain this, so I'll go the easy route. At the fundamental level of the structure of the universe, nothing is touching. Atoms are separated by large distances and the structures that make atoms are separated by large distances. If you were millions of times smaller than an electron, the distances would be incomprehensibly vast, from your point of view. But at our level, almost everything appears smooth (or connected). Look at a piece of metal or glass and it will appear smooth, but to the person that is millions of times smaller than an electron, the metal or glass will be full of large holes.

  With imagination, one can envision a smooth surface that is continuous (no gaps or holes). But there is no such thing in reality.

  Now, an electron is called a "fundamental particle", since it has no known structure. Would an electron have a smooth surface? Does "surface" even have any meaning when speaking of electrons? I don't know, but maybe someone here does.

   With mathematics, you can graph a continuous function, like f(x) = x. It'll just be a straight line, with no gaps or holes. But numbers, along with continuity, are imaginary.

  EDIT:

  I just thought of 2 things that challenge my notions:

  I've heard that neutron stars are made of "neutronium", which is densely packed neutrons. They might be touching. I don't know.

   Also, the interior of black holes is unknown. It might be a form of matter that is continuous, due to the nearly infinite density.

Please allow me to correct some of your misconceptions (don't hold it against me, I do this kind of stuff all the time   It seems to me like quite a popular thing these days is to "learn" physics, especially quantum and string theory.   In reality, in order to understand physics you have to take courses and truly understand everything from basic mechancis, to E&M, optics, waves, classical mechanics, statistical mechanics, atomic, and only then can you begin learning quantum... in other words, reading a few popular books doesn't cut it.  On the other hand, I think it's always good for everyone to keep learning, keep asking questions, and keep challenging oneself!  So it's good that you're thinking about such things.  Below are my notes for you.

1)It makes no sense to say "at the fundamental levels".  Levels are relative, therefore you have to have a frame of reference. 

2)What do you mean by approximating reality?  That also makes no sense. 

3)Smaller than photons?  Again, that has no meaning because photons are measured by their wavelengths and come in many sizes from tiny to huge.

4)What do you mean what we consider visible light?  Visible light is by definition visible light!  There is no "what we consider"!  I also "see" things all the time in the infrared, radio, X-ray, etc when I look at various astronomical images...

5)Once you get into small particles, you don't talk about them like everyday (macro) objects.  An electron is not defined by "surface", that has no meaning.  It is defined by it's spin, it's mass, and the probability of finding it.  That is all. 

cheers,

-avatar!

edit: clarified a sentence or two

To avatar!:  It's been a while, but I thought electrons were defined by four quantum numbers, including energy level (n), angular momentum (l), azimuth (m sub l) and spin (m sub s).  The Pauli exclusion principle then stated that no two electrons can have the same set of quantum numbers in an atom or molecular bond.

To Jodo Kast:  From what I recall from my QM classes, when you start to get into higher vibrational/rotational states for molecular bonds, the difference between energy levels of successive states collapses to essentially nothing, producing a continuum of energy levels.  At temperatures that we're accustomed to on Earth, everything is in a fairly high translational/rotational/vibrational state, so from an energy perspective, we can have continuity.

I definitely disagree that numbers are imaginary.  The number four is very real, and I can give a very good physical representation of the presence of fourness; I can have four coconuts, for example, or four rabbits' feet.  Even so-called imaginary numbers have very real applications in the real world (they make dealing with periodic functions a whole lot easier).

Here's a question to you:  suppose that you consider only the set of rational numbers.  Is it continuous?  Or is it discountinuous?  You could claim that it is discontinuous because it leaves out all the irrational numbers, but you can approximate any irrational number with a rational number with arbitrary precision (e.g., even though pi is not rational, I can come up with a rational approximation of pi that is correct to the 80,000,000th decimal place).  What about the set of all algebraic numbers?  Is it discontinuous because it leaves out all the transcendental numbers?  And is the set of transcendental numbers not continuous because it doesn't contain zero?

In my mind, continuity depends on both what you are measuring, and what you are using to measure it.  If you say that an atomic monolayer is continuous because there are no atoms missing from their predicted spots, fine.  If you say that the monolayer is discontinuous because one of the electrons is out of place, then you are imposing a necessary degree of precision upon this monolayer, and you should reference it as such.    As avatar! said, you need to choose a frame of reference.

Hey Crash,

I've never heard the term azimuth, but rather I've always heard it called the magnetic quantum number (m_l).  Along with the principle (n), orbital (l), and spin (m_s) they certainly are the quantum numbers that arise from the solution to the Schrodinger equation for the hydrogen atom, however that's not the same thing as an electron.  I think the way I defined electron for Jodo is fair, although it certainly is in layman's terms.  Still, you certainly can't argue that it's a lepton with energy mass 0.511 MeV. 

cheers,

-avatar!

After checking up, it looks like l is the azimuthal quantum number, and m sub l is the magnetic quantum number.  I guess I was getting my terms confused.  Then the quantum numbers define the spacial probability map for the electron.  So, yeah, it's not the same as an electron; it's just one of the properties that electrons have.  Gotcha.

1. To me, "fundamental levels" are the places where subatomic particles exist. I don't know how else to say it. We are at a different level and are affected by gravity. So, I suppose it's safe to say that "fundamental levels" are those places where the force of gravity can not penetrate. According to the physicist Penrose, electrons are not affected by gravity.

  2. You're right. It makes little sense. There is no agreed upon definition of reality. But believe me, I have some suspicions of my own and explaining them would be difficult. I would have to write a book, probably.

  3. I messed up here. Now I remember reading that atoms are smaller than the wavelength of visible light, not the photons themselves. How big are photons, anyway? I've read that they are massless, but I don't quite understand that concept.

  4. I mean with respect to the optical nerves within the eyes of human beings. In my opinion, visible light may be quite different, depending on the species. Alien species, that is. We haven't had the opportunity to visit other star systems and galaxies. Some aliens might see radio waves and not see what we consider "visible light". Some might even see gamma rays. We might want to avoid those guys (since gamma rays would not harm them).

  5. I am greatly confused as to how an electron can not have a surface. If it does not have definable boundaries, then what is it? Does it at least have a point where one can say "this is not part of the electron"? Without some sort of surface, it could extend out and out, indefinitely.

I don't care, as long as that Schrodinger guy stays the hell away from my cat!

No need to worry, even he can't come back from beyond the grave. Why couldn't he have used a dog as an example instead? Cats are way too lovely for such a cruel thing even though it was just a mind game. A dog would've been equally fine.

1)If Penrose really believes electrons are not affected by gravity, then he doesn't know what he's talking about!  Everything, including photons, are affected by gravity.  This forms the basis of general relativity.

3)The length of a photon is the wavelength of light.  Therefore visible light has a length between 400 - 700 nanometers.  Yes, photons are the only particles that we know for certain are massless.  There are however other particles, which if they exist, are likely massless as well (such as gravitons).

4)If we saw in IR, then our definition of visible light would change in that "visible" would now be longer than 700 nm, however that certainly wouldn't change the object itself.  In other words, just because some creatures see different wavelengths of light, that doesn't affect the photons nor their place of origin.

5)We don't discuss electrons in "common" terms.  Surface has no meaning as it does in everyday life.  You define the "size" of an electron by its shell, meaning that at a given point you have a certain probability of locating the electron.  It's hard to understand, but that's how it is!  The quantum world is strange, and there are certainly things we can't answer at the moment, and yet so far it has proven 100% accurate.

cheers,

-avatar!

edit: Added some info

Ah, I read the article again (June 2005 Discover magazine) and Penrose stated that most physicists ignore the gravitational effects of electrons and other particles since they're negligible. The article mentions several times that gravity barely affects those particles and I remembered it wrong, apparently. Blame confabulation. For some reason, I took that as "not affected by gravity". In fact, I remember what I did. I decided to make my own explanation of reality and reasoned that since electrons are barely affected by gravity, they might as well not be. Which explains why they can be in more than one place at the same time. I should've wrote that down in a journal or something.

The point of the article was to explain why electrons can be in more than one place at the same time. There are 2 main interpretations, the Copenhagen and "Many Worlds". According to Copenhagen, something doesn't occupy a definite state or location until it's measured. According to "many worlds", everything exists in every possible state, but there are infinitely many universes, so we can see never see something in 2 states at the same time, like a photon going through both slits at the same time. Penrose offers a 3rd explanation and it's in the article. It would take a lot of typing for me to get it all out, mainly because I don't entirely understand it yet. It has something to do with objects settling into states. The larger something is, the more quickly it can settle into an observable state. The smaller something is, the longer it takes. The article estimates it takes 1 second for a dust speck to collapse to one location and 1 trillionth-trillionth of a second for a human to collapse to one location. It doesn't mention how long it takes electrons, which makes me quite suspicious. Meaning I'm allowed to make my own explanations.

Mathematicians can come up with all kinds of crazy and ridiculous theories that are completely out of touch with reality.  Those above sure sound like some of them.  I wouldn't invest for than 10 seconds of your time in them, if I were you.  Let me tell you, anyone can come up with a theory and then say "you can't prove me wrong!"  which may very well be true.  However, if there's no way to test the theory and it's not based on any real physical observation (despite what Penrose tries to purport) then I say forget it.  Just my opinion of course!

cheers,

-avatar!

Got to say I agree with you there Avatar. Mathematicians will tell you that Maths is the language of the Universe and yet they seem to do an awful lot of messing about to make stuff fit - they change Maths to fit the Universe, rather than the Universe conforming to the rules of Maths.

Let me just say that, even on a really basic level, if you have to bring in imaginary numbers into it, you know things aren't going your way.

But then I dropped out of University Maths and Physics pretty much for those reasons so what do I know?

I'm curious. If you had to explain the quantum world, what would your interpretation be? It's been verified that electrons can be in more than one place at the same time and no one has been able to explain why. Just until recently, there were only 2 main explanations, the Copenhagen and "many worlds". All Einstein had to say about it was that "God does not play dice", so he didn't even want to mess with it.

  I would say that "many worlds" is most likely incorrect. The Copenhagen interpretation is accepted by most physicists, but it still seems ridiculous to me.

  Penrose asserts that gravity forces electrons to settle down to one location. I find that agreeable, but it doesn't explain why an electron can be in more than one place at the same time. And then we have what Einstein called "spooky action at a distance". I don't understand the physics of it, but I've read it's been verified that information can travel faster than light (even in a vacuum), which ties into the thread I made about teleportation. This also makes me wonder what information is. There's a lot to wonder about, actually.

There sure is lots to wonder about!  My view on the matter is that an electron is NOT at two places at the same time, rather that we can't take a quantum view into our current macro view.  This is really the same as the Copenhagen interpretation which states we should only discuss the statistical probabilities.  It does seem a bit of hand-waving, and many people think the theory is incomplete, however currently it's the only one I believe that actually makes sense. 

cheers,

-avatar!