Why the Ground State Ain't Zero

If you are familiar with Max Planck's work or Albert Einstein's photo-electric effect, then you know energy comes in discrete packets called quanta. These discrete energy levels are represented by nice, neat integers, where n = 1, 2, 3 ... But then there is this equation:

The above equation's second term is the ground state--but why isn't the frequency (f) and Planck's constant (h) multiplied by a nice, neat integer? Why 1/2? That's odd. And why ain't the ground state zero? Surely if you remove everything there should be nothing, nada, zip! Definitely not 1/2.

To understand what's going on, let's try a thought experiment. Imagine you want to push a couch a distance of x. You put your hands on the couch and apply zero force or energy. You gradually increase the energy applied until you reach a critical value where the couch begins to move. Let's label that critical value 1E.

Now suppose you want to push two couches along distance x. You gradually increase the applied energy to, say, 1.7E:

Unfortunately, 1.7E is not enough energy to do the work, so gradually increase the applied energy to 2E:

Assuming the couches are identical, the critical values of energy needed to push one or two couches are 1E and 2E. The critical coefficients are integers. If we want to move n couches we need nE. Below is a diagram of our thought experiment:

Note that the energy you applied is continuous, but the critical values (in red) are discrete. Also note the lowest energy is zero, so again, where does the (1/2)fh come from? Consider a single die. It has six discrete states: 1, 2, 3, 4, 5, 6. If we add up these states, we get 21:

This time, instead of a continuous line, the following die diagram has discrete steps:

Now, let's imagine a die with an infinite number of sides (or states) ranging from zero to six. Its states are no-longer discrete, but continuous. However, like the couch experiment, there are critical values marked in red (see diagram below).

Once again note the ground state is zero, but also note when we add up all those energies (see equation 3) we don't get 21 like before. We get 18! The following diagram illustrates the difference between the six-sided die and the infinite-sided die. The six-sided die clearly has more area under the curve. That extra area is indicated in gray:

So how do we fix this discrepancy? Let's include a y-intercept as they say in calculus parlance:

At equation 5 we use an intercept of 1/2. Doing so gives us a sum of 21--equal to the six-sided die! The diagram below illustrates this point. Note the gray areas cancelling each other:

Also note the ground state is not zero--it's 1/2.

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.