Monday, January 03, 2011

Is String Theory wrong?

In String Theory, one of the most crucial calculations involves summing all the numbers from 1 to infinity, which obviously should be infinite. But not in ST, where the Rieman Zeta Function is used, and gives a value of -1/12??? Yes, if you add all the numbers from 1 to a million, you get a big number; if you add all the numbers from 1 to a trillion, you get an even bigger sum. But should you add all of them to infinity, not only do you get a fraction, but a negative one?

Here's a short version of a proof (attributed to Euler):

Euler started with this:

(1) 1 + x + x2 + x3 + · · · =1/(1 − x)

He differentiated both sides:

(2) 1 + 2x + 3x2 + · · · =1/(1 − x)2

He set x = −1 and got this:

(3) 1 − 2 + 3 − 4 + · · · =1/4

Then Euler considered this function (now known as the Riemann Zeta function):

(4) ζ(s) = 1−s + 2−s + 3−s + 4−s + · · ·

He multiplied by 2−s:

(5) 2−s ζ(s) = 2−s + 4−s + 6−s + 8−s + · · ·

Then he subtracted twice the second equation from the first:

(6) (1 − 2·2−s) ζ(s) = 1−s − 2−s + 3−s − 4−s + ···

and setting s = −1, he got:

(7) −3(1 + 2 + 3 + 4 + · · · ) = 1 − 2 + 3 − 4 + · · ·

Since he already knew the right-hand side equals 1/4, he concluded:

(8) 1 + 2 + 3 + 4 + · · · = −1/12

So what is wrong in this deduction? Take equation (3).

(3) 1/4 = 1 − 2 + 3 − 4 + 5 - 6 + 7 - 8 · · ·

We can regroup it as (take the first two, then the next two, etc.):

(3a) 1/4 = − 1 − 1 − 1 − 1 · · ·

Or we can regroup (3) as (take the first and third, second and fourth, etc.):

(3b) 1/4 = 4 − 6 + 8 − 10 + 12 − 14 · · ·

Now regroup (3b) as:

(3c) 1/4 = − 2 − 2 − 2 − 2· · ·

Comparing (3a) and (3c), we get that:

(9) 1/2 = 1/4 ???

We can conclude that Euler's deduction was plain wrong.

In all the textbooks, you will find that equation (1) is true only if |x| < 1 or -1 < x < 1, so by setting s equal to -1 in equation (3), we have made equation (1) divergent.

Now in all String Theory textbooks, we are told that ζ(-1)= -1/12 can be derived vigorously.

Stay tuned.

EDIT: See: Zeta Function is OK