Where Are the Extra Dimensions Hiding?

Imagine a line x along a plane. How many such lines can the plane hold? An infinite number:

Imagine a plane within a cube. How many such planes can the cube hold? An infinite number:

Now this is harder: imagine a cube within a 4D space. How many cubes can the 4D space hold? An infinite number:

Thus there are an infinite amount of cubes or 3D spaces inside 4D. According to the equation above, this is true no matter how small wxyz is. We can infer that space that is 4D or higher can accommodate an infinite amount of 3D space. The above video shows Brian Greene talking about a little ant crawling around in a tiny, curled-up higher dimension. Here is an illustration:

According to Greene, the ant is able to enter and exit the higher dimension with ease. However, according to the math above, the ant would have to travel an infinite distance. It would never make the trip within its puny lifetime. Below, the red arrows represent the ant's journey. The vertical line represents 3D and the circle represents a higher curled-up dimension. Note the lines within the circle. They represent 3D spaces--an infinite number of them.

Now where there's infinite 3D space, there is bound to be infinite ground-state energy and mass:

So within a higher dimension (no matter how small) there should be infinite energy and mass! If these higher dimensions exist, every volume of stuff we measure should have infinite energy and mass! But they don't. That suggests strongly that higher dimensions are imaginary. There is also the Heisenberg Uncertainty principle which tells us that something very small with infinite energy should exist for zero amount of time:

Then again, what about the many many dimensions of Hilbert space? Where would quantum physics be without those extra dimensions? We should take a close look at Hilbert space. Let's start by examining a familiar 3D space, then we will analyze a 6D object I recently discovered.

Consider vector A below. It is composed of unit vectors i, j, and k:

Let's take the dot product of A with itself.

Taking the dot product of A with itself leads to the identity matrix or Kronecker delta. Because space dimensions are orthogonal (90 degrees to each other), the products of the cross terms equal zero. "Orthogonal" is a good thing--it suggests the 3D space is legit. If the dimensions are orthogonal, the dimensions should also be linearly independent. Let's check this. First we define the eigenvectors:

Next, we multiply each eigenvector by a coefficient (ci,cj,ck), then add them to get a column vector with all zeros.

It is clear that all the coefficients equal zero. This spells linear independence. Now, below is the math for the 6D object I mentioned earlier. As you can see, its dimensions are also orthogonal and linearly independent.

This 6D object does not have infinite mass or energy or infinite 3D space within. The 6D object is none other than a single die:

Notice that the unit vectors i and j are parallel lines; i.e., ijcos(0). Their product does not equal zero, so they are not orthogonal. When we test for linear independence, equation 19 reveals that ci does not necessarily equal zero, nor does cj. So the die's spacial dimensions are not 6D, but 3D.

So what are the die's six dimensions and why are they orthogonal and linearly independent? We can think of equation 20 below as the wave function of the die. At equation 21 we take the dot or inner product of the wave function. Notice at 22 the cross-term products are all zeros within the matrix. How is that possible? (See equations 23 to 25.)

Equation 25 reveals the 6D has nothing to do with space or its angles. It has to do with probabilities! The cross-term products are zero because there is a zero probability the die will yield any numbers that aren't 1, 2, 3, 4, 5, 6. Equations 26 and 27 provide an example of a cross-term product and its probability:

Thus the die's dimensions are orthogonal because they are each statistically independent--not because of right angles. However, we can say that p (the variable subsuming probabilities) is equivalent to cosine-phi for Cartesian unit vectors--at least for the cross-term products, since they are all zero.

This equivalence might lead one to believe there are six spacial dimensions, but what we really have are six statistically independent states within Hilbert space. We can determine the expectation value of these states as follows:

As you can see, having extra dimensions, parameters or states can be useful--and they can reside within 3D space.

Probabilities, Euler's Identity and Super-complex Numbers

Today we are going to examine how super-complex numbers fit in with Euler's identity and probability amplitudes. If you are not familiar with super-complex numbers, read my post entitled: "Introducing Super Complex Numbers."

Euler's identity is as follows:

It has a cosine and one isine or imaginary number. It can be used to model a right triangle:

Or a propagating wave:

Here is the super-complex number version of Euler's identity:

The exponent has i sub-n and theta sub-n. This tells us we will take the cosine of angle theta, and the isine-thetas from 1 to n. In the example below, n = 3.

The super-complex Euler identity is useful if you are working with one cosine and multiple triangles that share the same cosine. Take the exponent i sub-n, theta sub-n; it shows precisely how many isines/triangles are involved. The super-complex Euler identity is a real time saver. Instead of writing a big long string of trig functions we can write one simple exponent.

Since it can represent multiple triangles, it can be used to represent a wing design for a stealth aircraft:

OK, so that wasn't really a stealth aircraft--just more triangles, but you get the idea. Below are multiple waves propagating through space that are modeled by the super-complex Euler ID:

We know that if we multiply Euler's identity with its complex conjugate we get 1. We can also take the inner products of cosine and isine to get probabilities, but notice we are stuck with just two probabilities:

Using the super-complex Euler ID, is it possible to have as many probabilities as we like--and they all add up to one? To answer this question we need to derive the super-complex Euler ID. Start with the normal Euler ID, then split up the isin(theta) into smaller parts. Each part shall have a coefficient of epsilon:

If we examine a unit circle diagram and some trig relations, we discover that each epsilon sub-j isin(theta) equals isin(theta sub-j). We make a substitution:

If we equate i with each i sub-j and i sub-k, we assume the product of any two indexed i's is -1. Unfortunately this will give us non-zero cross product terms.

Let's check to see if the cross product of two arbitrary indexed i's really do give us -1 and not 0. Take the sum of two arbitrary terms and multiply it by its super-complex conjugate. That will give us a real number we label b^2. Like the isine of Euler's ID, we split b into smaller parts. Call those parts c and d.

The last equation above confirms the product of any two indexed i's equals -1. The great thing about Euler's ID is the square of its absolute value yields the same result as a dot product. We want the super-complex Euler's ID to behave the same way--no non-zero cross products! So here's what we do:

At the second equation above, we assume there are no non-zero cross product terms on the right side; however, without the extra cross-product terms, the left side is greater than the right side. We cure this by increasing the value of the isines. We then throw in unit vectors (ej). We now have a multiplication that behaves like an inner or dot product. We make some further refinements below:

Note how each indexed i is converted to an indexed mu, which behaves like a unit vector, giving the result we want:

Each squared sine can represent a probability and the total is 1.

As you can see, the super-complex version of Euler's identity has a great deal more flexibility than the ordinary Euler's identity. It allows us to model complex systems and probability amplitudes with just one simple exponent expression.

Introducing Super Complex Numbers

We know that i and -i are square roots of -1. They are part of the axis of imaginary numbers. Combine them with real numbers and you get complex numbers. Complex numbers are often used to model rotations, spins, oscillations, vibrations. They can even be used to model error margins. 55 +/- 3 can be expressed as 55 +/- 3i.

Complex numbers are useful any time you have a real-number value combined with a number that fluctuates. A good example is a wave. The real number tells you how long the wave is or how far it has moved along the x-axis. The imaginary number tells you the vertical measurement along the i axis. (See diagram below.)

Point A above is the sum of the real part and the imaginary part. Complex numbers work well in the above example, but suppose you want to model, say, an electric wave and a magnetic wave? For that you may want to use super complex numbers. Super complex numbers have an extra imaginary axis so you can model two waves for the price of one:

As you can see, i1 and i2 are both equivalent to i. They are both square roots of -1. Multiplications between them yield -1 or 1 in the same manner as plain old i. Operations of super complex numbers are similar to complex numbers.

Division with super complex numbers is tricky just as it is with complex numbers. You use super complex conjugates ( super complex numbers with the signs reversed) to get a real number solution:

The type of super complex number we've been working with so far is called a type-2 super complex number. It's type-2 because there are two imaginary axes. A complex number is a type-1 super complex number, since it only has one imaginary axis. Real numbers are type-0 for an obvious reason. All this implies we can have type-3 or even type-infinity super complex numbers.

Suppose we have multiple waves propagating through space? We can model the entire system with the following expression:

So far, all our waves have conveniently moved along the x-axis. What about the y and z axes? Or some combination of axes? Suppose we have waves moving along a vector? The imaginary axes would have to become imaginary vectors with the same angular relationship to the real-number vector as they had with the x-axis.

Below is a super complex vector expression:

But why stop at super complex vectors when we can have super complex tensors? The diagram below shows a type-n super complex tensor of rank-2. Below that is a general expression that can fit any super complex number or tensor.

Imagine being able to model a highly complex system filled with fixed values and variations. The weather perhaps? You could also model beams in a building as they vibrate during an earthquake. The beams could make up real-number tensors, and all the different vibrations could make up the imaginary-number tensors. These are but a couple of examples of what you can do with super complex numbers. Their application is only limited by your imagination.

For more information of this topic see "Probabilities, Euler's identity and Super-complex Numbers."

General Relativity's Invariant Tensor Myth

If you took a General Relativity course or read a book on the subject you were probably told the tensors that make up Einstein's field equations are a good thing because they are invariant. What's so special about invariant tensors? Here is a quote from one source:

"As an abstract mathematical entity, tensors have an existence independent of any coordinate system or frame of reference ..."

Wow! Cool! This means I can take a vector (a rank-one tensor), place it in any coordinate system or reference frame (also known as a basis) and it will not be affected by the coordinate system. The only things that will change are the values of the vector's components. The diagrams below show an example of a typical coordinate transformation:

Notice how the vector looks and behaves the same way in the different coordinate systems. However, Einstein's theory of General Relativity would not work if this were really true. If the vector above was a light beam this is what would happen:

The light-beam vector curves if the geometry of the coordinate system is curved. It is not independent of the coordinate system. So technically, it is not a tensor and should not be modeled by tensors. Or, the definition of "tensor" needs to be modified.

The theory of General Relativity claims it is the very geometry of spacetime that causes the light beam to curve. Also, curved spacetime geometry causes other objects that are considered tensors to move along a geodesic or curved trajectory when those objects would move differently in flat spacetime. None of this is consistent with the concept of invariance.