How to Derive M-Theory's Eleven Dimensions and Reduce Them to Four Dimensions

Today we are going to mathematically derive the eleven dimensions of M-theory. This is part three of a series of posts regarding the string theories. To fully understand what's going on here, I recommend you read "Debunking Bosonic String Theory's 26 Dimensions" and "Are String Theory's Extra Dimensions Real?"

Once again we have an x-y plane or system zooming along the z-axis with momentum p. The frequency of the ground-state oscillator is n/2. We want to add up all the values of n and get infinity minus one. At step one below we multiply the sum by the exponent e to the minus epsilon power--which is equivalent to multiplying by one. Epsilon is a very tiny number that is practically zero.

The sum is equivalent to minus one-half the derivative of the sum without multiplying by n.

Since the sum adds to infinity we can replace it with another expression that amounts to infinity: an exponent e divided by one minus the exponent e.

We then convert the expression's numerator and denominator into a Taylor expansion of exponent e, and then simplify.

Next, pull out one-over-epsilon from the fraction.

The variable s, like epsilon, is a tiny number. We can use the rule below to convert the denominator.

We can now multiply the numerator by the converted denominator. Note that the simplification leaves out the terms that cancel and have high powers of epsilon. These high-power terms drop to zero when epsilon goes to its zero limit.

We simplify further by multiplying the parenthetic terms by one-over-episilon, taking the derivative with respect to epsilon and multiplying by minus one-half.

We end up with infinity minus 1/8. We need a minus one, so we need to multiply -1/8 by eight. That gives us eight dimensions. Add to that the z-axis plus time--and we get a total of ten dimensions.

So then why does M-theory have eleven dimensions if there are only ten? Back in the 1990's there were five string theories. (Now there are around E500 string theories!) How could any of those theories be the unifying theory when there were five of them? M-theory to the rescue! According to M-theory, those five theories (or E500 theories) are just different forms of the same theory.

Imagine that each of the string theories describes a unique 10D universe. Imagine all those 10D universes existing in a higher dimension. Yes, the eleventh dimension. Think of the eleventh dimension as a line (or string if you prefer), and each 10D universe is a point along that line.

With a little math similar to what we were doing above, we can derive the eleventh dimension below. We take our infinity minus 1/8 (which includes the z-axis and time) and add up an infinite number of them.

We get infinity minus one. The minus one times one is one more dimension to add to our collection. That brings us up to 11D.

But as you probably know, we can also derive 26 dimensions. We can also derive four dimensions:

So how many dimensions does our universe have? Which mathematics is telling it like it is? Well, this is where our powers of observation come in handy. The most useful mathematical models are the ones that are consistent with the reality we observe. For now, it is prudent to go with 4D, since that is what we have observed.