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.

Are String Theory's Extra Dimensions Real?

Why do positrons and electrons have opposite charges? One explanation involves an extra dimension: the positron moves in one direction down that dimension and the electron moves in the opposite direction. The result is opposite charge. But you might ask, what prevents the electron and the positron from changing direction? Do positrons become electrons and vice versa? If they do, that would be a sign that the extra dimension exists. If not ... oh well.

If extra dimensions do exist, here is a scenario that should happen:

You have three hydrogen atoms. One has an infinite mass; the second one has the mass you expect; the third one has a mass of five trillion kilo-tons. How is that possible?

If you look carefully at the dimensions we are familiar with, you will notice a pattern. There are an infinite number of points along a line; an infinite number of lines along a plane; an infinite number of planes inside a cube. In short, a higher dimension can accommodate an infinite number of lower-dimensional objects. If this pattern is consistent, then a 4D space could accommodate an infinite number of 3D cubes. We can check this:

A cube is measured in cubic meters or m^3. A 4D hyper-cube is measured with m^4. How much 4D room does an m^3 cube take up? x * y * z * 0 = 0 m^4.

As you can see, the cube has magnitude along x, y, and z, but zero magnitude along the 4D axis. As a result, it takes up zero m^4 space. And, it can be any size! So an m^4 space can hold an infinite amount of 3D space or matter, or energy.

Now imagine there are extra dimensions in a tiny region of, say, a hydrogen atom. Particles are free to enter and exit this region. Once inside, the particles encounter an infinite 3D universe. So potentially, they could gather there and cause the hydrogen atom to have significantly increased mass and energy.

Ironically, the m^4 space could be zero and still hold and infinite amount of of 3D space, matter and energy, since anything 3D takes up zero m^4 space. So if extra dimensions exist, it does not matter how short or curled up they are. They will accommodate infinite 3D.

This presents a big problem for physics, since you could not count on two identical items having the same mass or energy. The fact that we can count on two identical items to be identical, is a sign that extra dimensions are highly improbable. More dimensions would destroy energy conservation, since up to an infinite amount of energy could leave or enter our universe at any time.

Are there other ways we can test whether extra dimensions exist?

Certainly! By definition, space dimensions are vectors that are orthogonal to each other; they are right angles to each other. So far we humans have discovered three: i, j, and k.

Suppose we think there might be a fourth dimension. Let’s call it “a.” We can test this dimension with cross products. We know that iXj = k; jXk = i; and kXi = j.

So the question becomes what is aXi? The answer has to be either j or k. If it’s j, then aXi = kXi, so “a” could be parallel to k. If it is not orthogonal to k, it is not really a fourth space dimension. If the answer is k, then “a” could be parallel to j, and our conclusion about “a” remains. aXj and aXk can yield more parallel vectors rather than orthogonal ones. It appears the cross product test falsifies the notion of a fourth spacial dimension or additional dimensions.

But suppose we change the rules. We decide, for example, that aXi = j or k. This is possible if the four dimensions are perpendicular to each other. However, physics would be less certain. We don’t know whether we will get j or k. Physics becomes even more uncertain when you add additional dimensions. If you have nine space dimensions, then aXi could equal any of the other seven vectors. Adding more dimensions to our physics does solve certain problems, but at the expense of creating new absurdities and uncertainties.

Some theoretical physicists want to discover extra dimensions so badly they can taste them--but only in their dreams. To date, there is no empirical evidence of extra spacial dimensions.