Season 17, Episode 58: Why Does String Theory Have So Many Dimensions?

Hey StarTalkians! Episode 58 of season 17 was another Cosmic Queries edition, with guest and string theorist Professor Lara Andersson, Neil and Jordan. Many of the questions concerned the – seemingly excessive – number of dimensions in string theory. This one really gets to the heart of the matter:

Quantum Anomalies with Lara Anderson - StarTalk Radio

(from 30:30)

Professor Andersson explains it very well in layman’s terms. Truthfully, it’s difficult to convey without a lot of mathematics, but here, I’ll try to address the core issue in a little more detail.

Why Does String Theory Have Extra Dimensions?

String theory is an attempt to unify all four fundamental forces – thus creating a “theory of everything.” When you’re doing this, it’s probably wise to not make any assumptions about the number of dimensions there are. After all, it’s possible that we aren’t aware of all of them.

So you can develop the ideas of string theory in a “dimension agnostic” way, just putting d in place of anything dependent on the number of dimensions.

Working this way, you eventually run into an issue. Any such theory would have to be invariant under a transformation called a Weyl transformation, but this can be broken by quantum mechanical effects. In order to form a consistent theory with these “Weyl anomalies,” they have to cancel out.

That’s when d comes back. It was found that with d = 26 (in the case of bosonic – force-related – string theory) or d = 10 (in the case of fermionic – matter related – superstring theory), you can ensure the cancellation. In this way, the dimensions kind of just come out of the theory, if it is to work at all.

How Many Extra Dimensions?

It depends on the specific theory. Five different variants of string theory quickly developed, which physicists found connections between by rescaling so-called “coupling constants” which appear in each. In 1995, Ed Witten realized that these five theories were just special cases of “M theory” – the mother of all string theories.

In M theory, there are 11 dimensions. This is therefore the best answer to how many dimensions are present in string theory.

So... Where Are They?  

As explained on the podcast, the complete lack of these dimensions in our experience is explained by “compactification.” This is questionable, from a philosophical perspective, but does fix the problem of “where are the extra dimensions?”.