Uncertainty Principle

There's a "Physics" joke. It goes:

An electron was stopped by a policeman for speeding. The Policeman says, " 'Ello, 'ello, 'ello. Do you know how fast you were going, sir?"

The electron answers "No, but I know where I am."

Heisenberg's Uncertainty Principle states that any item cannot simultaneously have both its position and its momentum (mass times speed) exactly defined. There is an inherent uncertainty.

Someone could give more details but there's a formula something like: (Uncertainty in position) x (uncertainty of momentum) = Planck's constant (called "h bar").

Heisenberg's principle is most applicable to very small bodies (electrons etc) because Planck's constant is very very small (If I remember right it's about 3.3 x 10 to the power -34).

This is pretty contentious. Heisenberg is saying just that this information does not exist. The world is uncertain. It totally contradicts the view of the world as blobs of solid matter doing solid kinds of things. It is the heart of quantum mechanics that matter is actually fuzzy and always moving around so that really things are not as certain as we intuitively believe.

This is all easier to understaind if you think not about whether the information is defined but about whether it can be known. Think about it in the way this joke puts it - it just says that you can't "know" both position and momentum at once.

Think about how you would measure the position of an electron... you would have to shine light at it to see it... But even light has some momentum and as the light was reflected back it would give the electron a kick thereby changing its momentum. Light is made up of little balls like photons. This would therefore be like throwing a number of tennis balls at a basketball to see where the basketball was at the start. By the time you had a good idea, the basketball would be on the move.

Similarly if you want to know an electron's momentum you have to measure its speed. To do this you have to look at it at a couple of different points in time. This inherently means you don't know its position and momentum (speed) at the same time - because you have measured it at two different places at two different times. Alternatively you could measure its position by simply stopping it. But by doing this you obviously change its momentum because its speed drops to zero. By stopping the electron, the policeman has changed its speed.

A wiki has this kind of uncertainty feature. When I read a page I never know if someone else is editing it as I speak. Also the very fact of me reading some pages makes me inclined to edit them.

But the above explanation works perfectly well in terms of classical physics rather than quantum, and does not yield any obvious absolute limits -- one can approximate some aspects of QM via "neoclassical analysis", taking into account lots of little things people didn't think much about in the 19th century, such as summation over many trajectories, chaotic trajectories, the effect of higher and lower wavelengths of observing light, etc. And that approach is interesting, but provides no assurance of whether there might be loopholes.

In quantum physics, on the other hand, the position and momentum of a particle are "adjoint", which means basically that they are both different aspects of the same quantum wave function, and gaining information via any means whatsoever influences the distribution of probability amplitudes in the wave function.

A exact measurement in one domain (e.g. momentum) is a multiplication by a Dirac Delta in that domain, which is a convolution with a non-compact distribution in the adjoint domain, which inherently (due to the purely mathematical Generalized Uncertainty Principle) causes it to lose any compact nature.

That is, it inherently (no possible loopholes) smears out the other variable completely over its infinite domain.

I don't think that this sort of harmonic analysis varies if the topology of space-time turns out not to be simply connected, but I'm not 100% sure. (I.e. for all I know, a wormhole connecting two regions of space might somehow change the math.)