String theory definitely spits out some strange and unbelievable things. Ten spacetime dimensions. Particles with negative mass. That sort of stuff. But usually there’s some justification for it. There will be a nice little section in the textbook outlining where the result comes from and what we can do to explain it.
However, about halfway through chapter 14 of Barton Zwiebach’s A First Course in String Theory, without any notice or fanfare, the author slips this little gem in on us (equation 14.84):
After some more algebra and problem solving the same thing shows up again, in more formal mathematical notation (equation 14.88):
Let’s just isolate the juicy bit here:
In plain english, the sum of all whole numbers from 1 to infinity equals negative one twelfth. I’ve tried telling some friends this and they all just stare blankly for a second and then say that that’s wrong. No question. It’s just wrong. I say I’m inclined to agree with them, but if it’s in a textbook it has to be right! You don’t just accidentally type “-1/12″ when you meant to type “infinity”.
What makes it worse is that I haven’t been able to find any reference to where this result comes from. There’s no explanation in the textbook, and the prof in class treated it similarly (“…and so that’s equal to the sum from one to infinity, but that’s just negative one over twelve…”), as if it were a regular everyday mathematical fact.
Would anybody please care to explain?