Measuring Fields Without Infinities

If we want to measure, say, the total energy of a field, it seems logical to measure the energy of each and every point in that field and add them all up to get the total. Unfortunately the field has an infinite number of points within its space, each with a finite amount of energy. The total energy is infinite according to our calculation--but when we actually measure the total energy, we get a finite value.

So let's forget about points in space. Instead, let's measure each wave frequency, then add them all up. Unfortunately, we get infinity again. We get infinity if we treat each particle-wave as though it had positive energy. Wouldn't it be great if a big chunk of the energy were negative? It would cancel the infinity and we would be left with the energy we actually measure.

Let's take a closer look at particle-waves. Below are a couple of waves. They each have a different frequency and amplitude. Note how each half cycle is either plus or minus, but not both; i.e., the plus and minus do not cancel each other. If we added the energies of these waves together, the total would be be the sum of their energies--or would it?

We know that each wave has its own wave function:

Let's do an experiment: Take a bunch of waves (up to an infinite number) with varied amplitudes and wavelengths, and run them all together:

When we looked at individual waves, we noticed that the plus half of the cycle never cancelled the minus half. The diagram above shows destructive interference when a bunch of waves are mixed together--the pluses cancel the minuses. Thus positive infinity is cancelled by negative infinity. Hopefully, we have something finite left over that agrees with actual measurements. Let's check and see.

We will now calculate the total energy in a vacuum. We will add up each integer (n) from minus infinity to plus infinity, and multiply the total by 1/2 the frequency (f) times Planck's constant (h).

Let's calculate the integral:

The constants (c) cancel. Next, we divide this definite integral into two parts: negative infinity to zero, and, zero to infinity.

This amounts to negative infinity squared plus positive infinity squared, which is equivalent to adding infinity to minus infinity--and the following equation where the epsilon limit is zero:

Substitute the Taylor series of each exponent and simplify:

As you can see, we end up with no infinities--just a finite energy.