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.

Debunking Bosonic String Theory's 26 Dimensions

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.

Why Too Few Dimensions are as Bad as Too Many

Zero-dimensional particles and one dimensional strings present a problem: They have a finite amount of energy but take up a zero volume of space, so it’s theoretically possible to have an infinite number of these particles or strings in the smallest space you can imagine. That implies infinite energy or a lot more energy than you expect to measure.

One hydrogen atom could have an infinite amount of energy if it contains an infinite number of strings. Another hydrogen atom could contain less than an infinite amount of strings, but still have far more energy than you would expect. A third hydrogen atom has the amount of energy you expect. This is the kind of crazy universe we should have if there are things that are less than 3D. As I showed in a previous post, you have similar problems when you have more than three dimensions.

Perhaps our universe really is 3D plus time, since that arrangement avoids the absurdities mentioned above.