>>10339676Okay, let me take you to my infinite featureless room of weirdness. I'll bring with me two identical coins. So I put one coin heads up on the floor and the other coin tails up on the floor somewhere else. From where you're standing, the coin to your left is heads. Marvel at my infinite featureless room that now has these two coins and nothing else interesting about it. Now is the part where I use the chloroform on you.
Some time later, you wake up. The coin to your left is now tails, and the other one is heads. Now, I SWEAR, I did not flip either coin. I'm a trustworthy guy, I wouldn't outright lie. So maybe you think I just picked them up and swapped them. But I am kind of an asshole who likes to trick you, as the chloroform demonstrates, so maybe I actually rotated the whole room 180 degrees underneath your unconscious body and didn't even TOUCH the coins. But you'll never be able to tell whether I rotated the room or just swapped the coins because the room's featureless besides the coins.
Now we just replace "coins" with "fields". I can get the same effect of manually swapping the two field operators by rotating them. But I know how fields rotate. Spin 1/2 particles need to rotate 720 degrees to be left unchanged; waving my hands magically and avoiding the details, 1/4 of a 720 degree rotation will give us a factor of i. We're swapping two fields, so rotating both gives two factors of i, leaving an overall negative sign.
