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Friday, November 13, 2015

Beyond Infinity and Quantum Mechanics

In another post having nothing to do with physics there's been a discussion of entangled particles, yet another non-intuitive aspect of quantum mechanics. First a joke:
Heisenberg is in the driver's seat, the officer asks "do you know how fast you were going?" Heisenberg replies, "No, but I know exactly where I am!" The officer looks at him confused and says "you were going 108 miles per hour!" Heisenberg throws his arms up and cries, "Great! Now I'm lost!" 
The officer, now more confused and frustrated orders the men outside of the car, and proceeds to inspect the vehicle. He opens the trunk and yells at the two men, "Hey! Did you guys know you have a dead cat back here?" Schrodinger angrily yells back, "We do now!"
I see from the web that not everyone gets the joke above, but I assume everyone reading this blog will.

When I've contemplated quantum mechanics and its decidedly non-intuitive nature (or lack thereof), I've pondered things where mathematics breaks down that have the same sort of feel and there's actually quite a few of them. I'm going to describe one now.

I'd like to propose a game. It's a wonderful game for you because you can only win. I'm going to write two checks to you. I'm not going to tell you the amounts of the checks except that one check is ten times the amount of the other check. I'm going to crumple up those checks and toss them into a hat and then shake the hat around. I'm going to then randomly pull the checks out of the hat and put one check in my left hand and one check in my right hand.

Here's the game. You get to choose either hand. I hand you the check in that hand. You look at the amount of the check. You can either keep that check or you can choose to exchange it for the other check. End of game. Simple, eh? And you're now richer, what could be better than that?

But there's something surprisingly spooky about this very, very simple game. Since it's completely random which hand the bigger check is in, it doesn't matter which hand you choose. Also, since it's completely random which hand the bigger check is in, there's no point in ever exchanging the first check for the one in the second hand, right? That would be just wasted effort, right? Wrong! (Sort of).

Let's take an example. You pick my left hand. I give you the check. You look at it. Let's say it's for $10. You now know that the other check is either for $100 or $1. The expected value of the other check is therefore ($100 + $1) / 2 or $50.50. $50.50 is much better than $10 so you of course (unless you're horribly risk adverse but let's ignore that possibility for now) choose to exchange the 1st check for the 2nd check. In fact, you would always choose to exchange the 1st check for the 2nd check. But that makes no sense, right? Because it's random, it shouldn't matter what hand you pick and the 2nd hand should be no better than the first!

Sort of like certain things in quantum mechanics. Measurements and observations affect the game. The above game is literally undefined until you look at the first check. The initial expected value of your winnings is undefined because it's a uniform distribution from zero to infinity and that distribution has an undefined expected value. Then, as soon as you look at the 1st check (but not until), the whole thing collapses and bang!, the game becomes defined and it always makes sense to trade for the 2nd check!

The game seems weird because it's not real. I can't actually write arbitrarily large checks. If there are limits, it actually makes sense that there might be a strategy that often involves taking the 2nd check.

The quantum is at the edge of reality and my guess it that it's in the realm where math breaks down, that it's in the realm that I call "beyond infinity." So nothing can be intuitive because we're so steeped in our every day math that when it doesn't work, such as in my game above, we're flummoxed.

Your probably wondering where your check is. It's in the mail of course!


Clovis e Adri said...

Good point, Bret. You pinpointed one of the main sources of our befuddlement over quantum mechanics: its probabilistic nature, always rendering situations analogous to your game.

The other source is its ondulatory nature, i.e. the fact it follows equations alike to the ones followed by waves, which renders also non-intuitive behavior for people coming from a mindset of classical particles.

And the two non-intuitive things above mold into one final super-non-intuitive thing: while any quantum systems evolves in absolutely deterministic ways (even though if wavelike), the moment you measure it, the probabilistic character of that measurement turns it into the spooky thing it is.

All that said, I wouldn't describe any of those things as being beyond mathematics. Quite the contrary, I am eternally amazed on how mathematics renders itself so powerful to describe our world...

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