About one hour into the above video, Leonard Susskind (one of my favorite professors) shows how string theorists mathematically derived 26 dimensions. Are you thinking what I'm thinking? If there are really 26 dimensions, then why do most string theories have fewer dimensions? This question made me watch the video with a CSI-investigator disposition.
The original goal was to get a minus-one ground state, so when a creation operator is applied, the photon will have zero mass. Imagine an x-y plane or system going down the z axis with momentum p. The frequency of the ground-state oscillator is n/2. Here is the desired equation and result:
If the result is infinity plus -1, that's OK, according to Susskind, it's just absorbed into the momentum along z and can be ignored. After some fancy math, including a Taylor expansion, the final result is ...
According to Susskind, if we drop the infinity (that lop-sided 8) we end up with -1/24. We need -1, so multiply by 24--that gives us 24 dimensions. Add the z axis and time, and that brings us to 26 dimensions. Voila!
But here's the rub: What if the result was the desired -1? How many dimensions would there be then? Let's see ... multiply by 1. That gives us one dimension. Add the z-axis and time--that brings us to two space dimensions and one time dimension!
Obviously the math and/or logic is seriously flawed.
Since it is OK to do away with infinities, we could take another approach. If you stop and think about it, we ignore infinities whenever we measure anything. How long is a meter? It's an infinite number of points. How big is a circle? It contains and infinite number of points. Everything we measure would be infinite if we didn't invent arbitrary units of measurement. We can do that here:
So instead of having 1/2 of an infinity, why not have 1/2 of a unit of ground state? And, while we're making changes, why do the frequencies of the ground state oscillators have to be a positive number when the ground state is negative? Why not be consistent? You'll note I made both sides of the above equation negative. The result is -1/2. Multiply by 2, add z and time--and voila! We live in a universe with three space and one time dimension after all!
Update: Here is another way to arrive at four dimensions instead of 26. It's based on the premise that the desired result is infinity plus -1/2. This ensures that within the infinity, there is a negative ground state.
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