Most physics professors I've spoken to don't actually understand things like quantum entanglement. They just say "the math works. that is all."
There is no intuitive understanding of a subject like that because neither 100K years of evolution of our brain now decades of real world experience expose us to it.
Seemed more than just lacking intuitive explanations. In entanglement for example, the answer is because it can't transfer information & you learn that just fine in any intro to quantum class.
It's not exactly covered convincingly though. The reason it supposedly can't transfer information is because you'd have to measure it, and we assume that measurement strictly follows Born's rules, which is a whole other can of worms. Sometimes people talk about density matrices instead, but we only use those since they are consistent with Born's rules. Additionally, many physicists point out that what you seem to be doing when you perform a measurement is entangling the system with the many bazillion degrees of freedom in the measurement apparatus and the world at large. We don't know, exactly, where the separation between a 'measurement-like' physical process occurs, or even if there really is one.
There isn't even a standard measure of entanglement for three or more particles. It's a pretty big mystery, still.
er, ok then i'll rephrase - "we're mostly sure it can't transfer.." or "we've never been able to get it to"
I doubt the writer meant it the same way you do. He was talking about the known things that hes not wrapped his mind around. In this case, explaining why the theories don't necessarily conflict should do.
The physical mechanism is that quantum mechanics governs a probability distribution (the "wave" in "wave-particle duality") on configuration space, not "physical" space.
Imagine you have two particles in 1 space dimension. The first particle has a position x (1 real number), the second particle has a position y (1 real number). "Physical space" would be the set of real numbers (R^1), while configuration space is the set of pairs of real numbers (x,y) (this set is R^2). The first number in the pair represents the position of particle 1, the second the position of particle 2.
An entangled state is simply a probability distribution on R^2 which is not a product state (i.e. P(x,y) != P_1(x) P_2(y)). This means that knowledge of the position of particle 1 gives you (possibly incomplete) information about the position of particle 2.
One example of this is "probability 1/2 that x=0, y=0, 1/2 that x=1, y=1". The minute you know x, you know y.
That's what entanglement is. What makes it weird is the idea that "knowing x" (measuring it) changes things. An explanation of this is trickier. Still understandable in principle (not just 'follow the math'), but too long for this post.
Ahhh... you just caused a major flashback in my memory. I remember reading about Banach-Tarski paradox about 7 years ago or so and it gave me so many headaches. I think I read the paper explaining it trice and it still made no sense to me and on my second reading I was convinced there was an error in an equation somewhere.. I asked a math prof who's a brilliant theoretician and he explained it to me and it still made little sense. Then I read von Neumann's explanation and it finally started making some sense. The key to understanding it is, like with many other concepts, to ignore your intuition and trust the math behind it.
I'd say trust your intuition more than the math, but I'm more physicist than mathematician.
To prove Banach-Tarski, you need to assume Zermelo-Frankel set theory (just ordinary set theory) plus the uncountable axiom of choice.
In my view, this means the uncountable axiom of choice is false. This actually breaks far less than you would expect. For most practical purposes, you only need the countable axiom of choice, which is the intuitively reasonable version.
The main thing the loss of the uncountable version breaks is the Hahn-Banach theorem on spaces of dimension higher than countable infinity. This in turn breaks quantum field theory on free space (not in a box), but that's broken anyway.
I personally thought that the Banach-Tarski theorem was a mess, and these non-measurable sets had to go. So I started studying nonstandard-analysis: analysis with infinities and infinitesimals as genuine, arithmetically viable, quantities.
Imagine my distress: the same problem occurs, Banach-Tarski still holds, unless you go into the rather more weird, Smooth Infinitesimal Analysis or Synthetic Differential Geometry; basically, treating geometry as made up of little discrete infinitesimally small points.
Reality can't be wrong, only our understanding or modeling of it can be wrong. So a paradox is either:
- our "intuitive" mental model of the world which says one thing can't become two of the same thing without using any more material, is wrong
- the mathematical model that says you can do that, is wrong
- the mathematical framework on which the model is built, is limited.
In other words, while I can't follow the maths, I would have no problem believing the mathematical weirdness is self consistent because I don't believe it would have any rational link to the real world anyway.
Your last paragraph seems to imply that you believe that somebody, somewhere is claiming this result is "real". In fact, nobody claims the theorem has any applicability to "reality", because the real world is quantized. You can't cut an electron in half. (So far as we know.)
Everybody knows the theorem isn't real. Now, I'm not just being pedantic here, because the fundamental and obvious failure of the theorem to have any connection to the real world is at the heart of one of the philosophical questions about the nature of math it engenders, along with its good friend the Axiom of Choice. While there's nothing wrong with either thing mathematically, does it really matter when anything that invokes the Axiom of Choice basically just opted out of concrete reality entirely?
Basically, I'm saying that the contents of your post are true, but trivial; there's no interesting argument about correspondence to reality to be made. The fun lies at the next level. I have no easy answers for you there.
(Personally, I have no problem breaking math into "Things that Might Be Practically Useful" and "Things to Entertain Mathematicians", especially since the latter tends to migrate to the former at quite surprising times, but many people seem to have trouble with giving such a rich meaning to the single word "math".)
It doesn't matter a jot when something "just opted out of concrete reality entirely", but it's also not especially interesting. Talking about pink elephants and how they can double in size may be internally consistent in a story, but it's not something that's "true but really difficult to believe in" anymore than "dividing by zero equals infinity" is difficult to believe in.
If the fun is at the next level, perhaps that's where I ought to go...
Thinking about it a bit more, that a mathematical model created by humans on a mathematical framework created by humans comes out with weird results sometimes - that fact is not even slightly weird or difficult to believe.
That such a human model can be as accurate and precise as quantum mechanics is, now that's hard to believe!
The Nonexistence one is what often gets me. If and when I ever give it more than a passing moment's thought, I have to take care lest it put me into an unsettling trance-like state of bewildered and bemused incomprehension and planar disorientation.
Saves money on jazzy cigarettes let me tells you.
I can relate to that. I used to feel a bit of a high when I was a young kid just trying to contemplate what would exist if nothing existed. Not nothing as in no matter, but no dimensions or time either. Thinking about that still puts my mind into a weird state.
Some people, including (I suspect) most theists, seem terrified by this, whereas I find it exhilarating and immensely humbling.
Funnily enough, it was actually while I was sat in church with my family when I was younger (I've since lapsed profoundly and comprehensively, of course) that my mind first began to wander to and wrestle with such concepts.
A better example is the well-ordering of the reals. From wikipedia:
From the [axioms of math] one can show that there is a well-order of the reals; it is also possible to show that [those axioms] alone are not sufficient to prove the existence of a definable (by a formula) well-order of the reals.
Absolutely nothing like the proposition that god exists, and I wag my finger at you for associating the concrete facts of mathematics with superstitious claims backed up by hearsay.
A well ordering is just some way of ordering a set made formal. Basically, if you have a set, you can find a relation such that any subset will have a least element. I should also point out that the example given of a set that is not well-ordered is wrong. The proof was found to be erroneous.
