An Incredibly Simple Formula For Removing Unwanted Infinities

Reality says, "It's finite," but your math says,"It's infinite." Why? And, is there a simple way to solve the problem? Today we present an incredibly simple formula for removing unwanted infinities. First, let's define the variables we will use:

Now that we got that out of the way, let's examine why infinities occur. One way to get infinity is to add up an infinite number of numbers as the following integral-summation shows:

You may have noticed something missing in equation 1 above. The thing that's missing makes the infinity finite:

The missing ingredient is none other than dx -- a really tiny number. When the infinity is multiplied by dx it magically becomes a finite number.

So one way to rid ourselves of an unwanted infinity is to multiply it by dx or something equivalent. Equation 5 below is a general formula that always provides a finite value by canceling infinity should one arise.

The first factor is equivalent to dx. The second factor could be equivalent to an infinite sum. Let's apply equation 5 to a classical example. Suppose we have a lattice with N cells. Each lattice cell contains some fraction (epsilon) of the total energy (E):

We could just add up all the parts to get the total energy (E):

We could calculate the average energy per cell (bar-epsilon*E) and multiply that value by the number of cells (N) to get the total energy:

But let's see what happens when we apply our new formula:

At equations 13 and 14, the dx equivalent is simply 1 which is often true in a classical case. It's a bit redundant but yields the right answer as do the more familiar methods above. In this example, it's like counting bricks to get the total. Very simple but effective. It's effective because the average energy per cell is simply the total energy divided by N--the number of cells. Thus, the bar-epsilon in our formula is the reciprocal of N--and the two variables cancel, leaving the total energy E. But look what happens when bar-epsilon doesn't equal the reciprocal of N:

In the above example we have infinite or infinitely uncertain energy in each cell, but the bar-epsilons and N's cancel, the uncertainty and/or infinity is nullified and we end up with the finite energy E.

The following example shows how infinity can rear its ugly head in a seemingly classical lattice.

Here's a data table and chart for different values of n:

Notice how the density increases as n decreases. As n increases the density levels off. The same is true for a 3D lattice:

When n is increased to infinity, the density is virtually constant and behaves like bricks. Increasing and decreasing the number of bricks does not change their mass density.

When n is reduced to zero, the density blows up to infinity! No more brick-density behavior:

We can translate the lattice equation into something resembling perturbation theory:

At equation 29 above we take epsilon to infinity (which is equivalent to taking n to zero) and the answer is 1 + infinity. We can use a famous ad hoc method to rid ourselves of the infinity: simply subtract it or ignore it. By pretending it isn't there, we get a legitimate finite energy value. On the other hand, it would make more mathematical sense to reduce epsilon to zero:

Let's apply our new formula to this problem. We begin with the original lattice equation and reduce n to zero to get the infinite result. At equation 33 we multiply the infinite density by the lattice volume to get the total energy--which is also infinite.

At 33b we restate the formula for convenience. At 34 we plug in the values we get from equation 33. At 35 we see the total energy is no-longer infinite, but the correct finite value.

Now, let's consider an even more classical setup. Here's the lattice diagram:

Surely the density should always behave like bricks, since we have one dot per cell. Sixteen dots per sixteen cells has the same density as one dot per one cell:

But even bricks have a limit. If you divide a brick up into smaller and smaller pieces, eventually you end up with something that no-longer resembles a brick. We can simulate this by reducing n to 1, and k to 0 (see 39 and 40). Once again we get the dreaded infinity--but we can fix that using the new formula (see 41 through 44).

How about a real-world example? At 45 below we set the total energy of two electrons equal to their Coulomb energy at a fixed radius (ro). You will note the total energy (2Ee) is quantized at a fixed integer value (no).

If the radius were to shrink to zero, one would think the energy would explode to infinity. To solve this problem we do some algebra to derive equation 52 below:

At 51 and 52 we see the radius cancelling itself. The total energy of the electrons is determined by the quantum number n. Apparently it is possible to have zero to infinite Coulomb energy (or force) without impacting the overall energy? The crude diagrams below sheds some light on this:

The diagrams show two Coulomb energies: E1 and E2 with radii of r1 and r2, respectively. There is a unit area A. Over a large surface area there are more units of A and vice versa. There are two circles; each representing two different surface areas. Using this information we derive equation 58:

Equation 58 tells us that a large Coulomb energy over a small surface area is equal to a small Coulomb energy over a large surface area, and the total energy is the Coulomb energy over unit area A times the surface area. The total energy is, of course, the total energy of the two electrons--and that energy is finite and conserved! This result is consistent with our formula:

If we plug in the following values and do the math, we get the finite energy of the two electrons:

Thus, we now have an incredibly simple and redundant formula that shows the connection between the parts and the whole at the classical and quantum levels.

Update: Below is a proof of the formula: