Each line has an infinite number of points. We tame this infinity by creating an arbitrary finite unit. For example, take the set of real numbers. Between 0 and 1 there are an infinite number of numbers:
Normally, we count using integers: 1, 2, 3, etc. But we don't have to do it that way. We could count like this: 1-infinity, 2-infinity, 3-infinity--all the way up to infinity-infinity. If 3 is greater than 2, than 3-infinity is greater than 2-infinity. So what we have are different magnitudes of infinity that make up our finite numbers. With this in mind, it seems reasonable to assume we could add up an infinite set of numbers and get a finite number. For example, we could take the entire line of positive real numbers ...
...and shape it into a circle:
Now infinity is equal to zero and 2pi radians, i.e., finite numbers.
Let's imagine we are extremely naive, we don't know the first integer greater than zero. So we decide to add up all the real numbers from zero to the next integer point. That gives us an infinity:
The vertical lines represent the infinite quantity of real numbers between zero and the question mark. They make a nice 2D drawing of a triangle. The average real number is at the half-way point. If we take this number (.5) and multiply it by 2, we get the right answer: not infinity, but 1. This is the basic logic behind n-space. We take an infinite number of points in space of any number of dimensions and map them to a 2D space. The average value (expectation value) becomes our vertical axis. We multiply this value by the horizontal axis to get the total area which is a finite value.
The above diagram shows how each point in the original lattice space is mapped to n-space. Each point in the original space becomes a vertical line in n-space. So an infinite number of points, lines, planes or cubes (lattice cells) become an infinite number of vertical lines. The average vertical line (bar-np) is multiplied by the horizontal line (nx) to get the area--which is the correct finite answer.
Why is the n-space area the correct answer--and not infinity? Consider the following diagram:
Max Planck found that if he added up a set of finite discrete energies, he got the correct finite value. The above diagram shows we can also add up an infinite set of continuous energies and get the same finite value! Whether the energies are discrete or continuous, the area under the curve is the same. Thus, finding the area under the n-space curve is a way to find the correct answer. (Take note that, throughout this post, we take the energy term normally reserved for a single particle and use it to represent any energy. Sometimes the frequency and Planck's constant are set to one.)
The following relationships show us how to get the values in n-space we need to calculate the correct, finite answer:
Now, we want n-space to help us solve infinity problems in the quantum as well as the classical realm. This is why n-space was derived from Heisenberg's uncertainty principle. Here are the variables involved:
Here is the derivation:
Equation 12 shows the n-space area is always greater than or equal to 1/2--or the ground-state:
According to equation 13, the total energy in a system, like Planck's constant, has two components, dimensions, or factors (nx, np). The horizontal dimension (nx) is derived from position, and the vertical dimension is derived from momentum. The total energy is equal to or greater than the ground-state energy. Using equation 12 we can derive equation 14:
At 14, k is a constant, so if equation 14 represents the total energy in the system, that energy is conserved. It does not matter how big or small the average energy is at any given point in the original space or lattice. Nor does it matter if there are an infinite number of such points. That energy or quantum number (np) is offset by quantum number (nx), giving the conserved quantity.
Now that we've laid the groundwork for n-space, let's attempt to solve a classic problem: calculating the total energy in a sphere, where each point in that sphere has a given amount of energy, momentum, and/or mass. And, of course, there are an infinite number of points in the sphere.
Immediately we run into a problem: if we know the exact energy at each point, we know the exact momentum if we divide the energy by c (light speed), and, it is obvious we know the exact position as well--a clear violation of the uncertainty principle. If we zero in on a point in space, according to Heisenberg, we should be totally uncertain about the energy and momentum. According to the de Broglie wavelength formula, if we reduce a wavelength to zero, i.e., a single point, we should have infinite energy! And, we can only know that if we have no clue where that point is located!
Below is the relevant math:
Realistically, each point of energy is not a point at all, but a wavelength with a one-dimensional magnitude. If the average wavelength is greater than zero, then we have a finite energy at each wavelength.
We can think of each wavelength as a line. Assuming we know the energies and momentums, we don't know the positions, but we can make this fact unimportant if we calculate the average energy/momentum. Then we know that at any randomly chosen position the average energy is always the same. We can then map each energy/momentum to n-space.
Using equations 21 and 22 we find the average vertical factor (np).
We use the following equations to find the horizontal factor (nx):
At equation 23 we see a problem. To find nx we must first know nt--the total that we are trying to calculate! So we move on to equation 24. We know the total volume but we don't know this thing called the unit volume. We get a unit volume by taking the volume of another system, where we know all the variable values, and multiplying that volume by a factor of np/nt (see equation 25). Once we have our unit volume, we can plug that into equation 24 to get the nx value for the instant problem.
We do the final steps below:
The strategy we used works as long as the following is true:
Suppose we have a scenario where we have a volume of energy, say, a star. The energy is conserved as follows:
The star collapses into a black hole. All wavelengths allegedly shrink to a zero limit. That forces the average momentum factor np to blow up to infinity:
At equation 31, the star's radius also shrinks to a zero limit. We should end up with a singularity that has a position unknown to us, assuming we know the total energy, mass, and momentum. We can imagine the singularity being anywhere within the Schwarzschild radius. Nevertheless, we can crudely map the star to n-space as follows:
As explained earlier, the positions of the wavelengths and the position of the singularity become irrelevant when we determine the average np for each wavelength. Now, let's assume we don't know the star's total energy. We want to find it, so we need to find the value of nx. The star volume is its radius cubed times 4/3 pi. The Schwarzschild radius cubed times 4/3 pi shall serve as the unit volume. We divide the volume by the unit volume--the 4/3 pi's cancel:
When we do the math we see that the star's total energy is finite and is equal to the black hole's (assuming energy is constant and none was transferred).
So in the case of the black hole, np had an infinite limit, but nx had a zero limit--so the total energy ended up being finite and conserved.